Prime Time | CBSE Class 6 Maths Notes

This note covers common multiples and the idli-vada game, factors and common factors in Jump Jackpot, perfect numbers, prime and composite numbers, the Sieve of Eratosthenes, patterns among primes, co-prime numbers and thread art, prime factorisation, divisibility tests for 10, 5, 2, 4 and 8, special numbers and the prime puzzle.

How does the idli-vada game introduce common multiples?

In the idli-vada game, children sit in a circle and count aloud in turn from 1. A player says “idli” in place of a multiple of 3, “vada” in place of a multiple of 5, and “idli-vada” in place of a multiple of both.

A player who makes a mistake is out, and the rounds continue until one player remains.

A multiple of a number is obtained by multiplying it by 1, 2, 3 and so on. Multiples of 3 are 3, 6, 9, 12, 15, … and multiples of 5 are 5, 10, 15, 20, … . The number 15 is in both lists, so it is the first idli-vada number.

Definition: A common multiple of two numbers is a multiple of each of them. The common multiples of 3 and 5 are 15, 30, 45, 60, … .

Two overlapping green circles are labelled “Multiples of 3” and “Multiples of 5”. The left part shows 3, 9, 12, 18, 21, 24 and 27 (the printed figure leaves out 6); the right part shows 5, 10, 20 and 25. The shared middle shows 15 and 30, marked as common multiples of 3 and 5.

See Fig. 5.1 in your NCERT textbook

Worked example 1. At what number is “idli-vada” said for the 10th time?

Answer: Idli-vada is said at the multiples of 15, so the 10th time is at 10 × 15 = 150.

How often is each word said?

Up to 90, there are 90 ÷ 3 = 30 multiples of 3, 90 ÷ 5 = 18 multiples of 5 and 90 ÷ 15 = 6 common multiples. Up to 900, each count is ten times larger.

Figure 5.1 matches the game: each circle holds the numbers for one word, and the overlap holds the idli-vada numbers. For a game up to 60, the overlap holds 15, 30, 45 and 60. The right part holds 5, 10, 20, 25, 35, 40, 50 and 55, and the left part holds the remaining multiples of 3, from 3 to 57.

What happens with other pairs of numbers?

For a new pair, “idli” goes with the smaller number and “vada” with the larger. The table lists the turns up to 60.

In one game, one number was 4 and nobody said just “vada”. Then every multiple of the vada number must also be a multiple of the idli number. From 2, 3, 5, 8 and 10, the number 8 fits, since every multiple of 8 is a multiple of 4.

The number 2 also fits: 4 becomes the vada number, and every multiple of 4 is even. With 3, 5 or 10, the larger number is itself said as just “vada”. The answer key gives only 8, which is the answer when 4 is the idli number.

How does Jump Jackpot connect jump sizes with factors?

In Jump Jackpot, Grumpy puts a treasure on a number. Jumpy picks one jump size and, starting at 0, lands only on its multiples. With the treasure on 24, a jump of 4 lands on 4, 8, 12, 16, 20 and 24. Jumps of 1, 2, 3, 6, 8, 12 and 24 also succeed.

Definition: A factor, or divisor, of a number divides it exactly, leaving no remainder. For example, 4 is a factor of 12, since 12 ÷ 4 = 3. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

A number line runs from 0 to 24. Dotted arcs above it show jumps of 2 (purple), 3 (orange) and 4 (black); dotted arcs below show jumps of 6 (green), 8 (brown) and 12 (red). Every set of jumps ends on 24.

Reference: NCERT Class 6, page 109, unnumbered illustration

Next, Grumpy hides treasures on 14 and 36, and Jumpy must reach both with one jump size. A jump of 7 lands on 7, 14, 21, 28, 35, 42, …, so it reaches 14 but misses 36.

Property: Jump sizes that reach two treasures are common factors

The factors of 14 are 1, 2, 7 and 14. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. Only 1 and 2 are in both lists. A number that is a factor of each of two numbers is a common factor of them.

Worked example 2. Which jump sizes reach both 15 and 30?

Answer: The factors of 15 are 1, 3, 5 and 15, and each of these also divides 30. So the jump sizes 1, 3, 5 and 15 reach both treasures.

What do the shaded and circled numbers show?

In a table of 31 to 70, the shaded numbers (33, 36, …, 69) are the multiples of 3, and the circled numbers (32, 36, …, 68) are the multiples of 4. Those both shaded and circled, 36, 48 and 60, are common multiples of 3 and 4.

A four-row grid shows 31 to 70. Some numbers are shaded green and some are circled in blue.

Reference: NCERT Class 6, page 110, unnumbered illustration

How are the first Figure it Out problems on multiples and factors solved?

• Q1. Counting in forties gives the multiples of 40 between 310 and 410: 320, 360 and 400.
• Q2a. Numbers below 40 with factor 7 are 7, 14, 21, 28 and 35, with digit sums 7, 5, 3, 10 and 8. The number is 35.
• Q2b. Numbers below 100 with factors 3 and 5 are 15, 30, 45, 60, 75 and 90. Only 45 has one digit 1 more than the other.
• Q4. 20 and 28 share 1, 2 and 4; 35 and 50 share 1 and 5; 4, 8 and 12 share 1, 2 and 4; 5, 15 and 25 share 1 and 5.
• Q5. Odd multiples of 25 are not multiples of 50, for example 25, 75 and 125.
• Q6. Two numbers below 10 whose first common multiple is above 50 could be 7 and 8 (56), 7 and 9 (63) or 8 and 9 (72).
• Q7. Treasures on 28 and 70 are both reached by the common factors 1, 2, 7 and 14.
• Q8. Guna's diagram keeps only the common multiples 24, 48 and 72, so the first common multiple is 24: multiples of 3 and 8 fit, as do 6 and 8, or 8 and 12. For 3 and 8, the circles hold 3, 6, 9, … and 8, 16, 32, … . The answer key's 3 and 4 fits only if the diagram shows just some of the common multiples, since 12, 36 and 60, also common multiples of 3 and 4, are missing.

What is a perfect number?

A perfect number is one whose factors add up to twice the number. The factors of 28 are 1, 2, 4, 7, 14 and 28, with sum 56 = 2 × 28. For Q3, the perfect number between 1 and 10 is 6, since 1 + 2 + 3 + 6 = 12 = 2 × 6.

How is the smallest multiple of many numbers found?

Worked example 3. Find the smallest number that is a multiple of every number from 1 to 10 except 7 (Q9), and then of every number from 1 to 10 (Q10).

Answer: A multiple of 8 is a multiple of 2 and 4, and a multiple of 9 is a multiple of 3. The first common multiple of 8 and 9 is 72, also a multiple of 6. Among 72, 144, 216, 288, 360, the first ending in 0 or 5 is 360, so Q9 gives 360. Among 360, 720, …, 2520, the first multiple of 7 is 2520, so Q10 gives 2520.

What are prime and composite numbers?

Guna and Anshu pack figs (anjeer) from their farm, 12 per box for Guna and 7 for Anshu, arranged in rectangles. In each of Guna's arrangements, such as two columns of 6, rows times columns gives 12. So the rows and columns are factors of 12.

Orange squares show 12 figs as one column of 12, two columns of 6 and three columns of 4. A single row of 7 purple squares shows the only arrangement of 7 figs.

Reference: NCERT Class 6, page 112, unnumbered illustration

Twelve has more than two factors (1, 2, 3, 4, 6 and 12), so it forms several rectangles. Seven has only two factors, 1 and 7, so it forms just one.

Definition: A prime number, or prime, has exactly two factors: 1 and the number itself. The first few primes are 2, 3, 5, 7, 11, 13, 17 and 19.

Definition: A composite number has more than two factors, so it has a factor other than 1 and itself. The first few are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18 and 20.

Property: A prime number has exactly two factors

A composite number always has an extra factor: 8 has the factor 4, and 9 has the factor 3. The number 1 is neither prime nor composite, because its only factor is 1.

Worked example 4. How many prime numbers and how many composite numbers are there from 21 to 30?

Answer: Only 23 and 29 have no factor other than 1 and themselves, so there are 2 primes. The other 8 numbers, 21, 22, 24, 25, 26, 27, 28 and 30, are composite.

How does the Sieve of Eratosthenes list the primes?

The Sieve of Eratosthenes lists every prime from 1 to 100 by removing non-primes in stages from a 10 by 10 grid.

1. Cross out 1, since it is neither prime nor composite.
2. Circle 2, then cross out every later multiple of 2: 4, 6, 8, and so on.
3. The next number not crossed out is 3. Circle it and cross out every later multiple of 3: 6, 9, 12, and so on.
4. The next number not crossed out is 5. Circle it and cross out every later multiple of 5: 10, 15, 20, and so on.
5. Keep going until each number is either circled or crossed out.

Every circled number is a prime, and every crossed-out number except 1 is composite. The procedure also works beyond 100.

A 10 by 10 grid holds 1 to 100. The primes are circled in green and every other number is crossed out in red. Below it, Guna and Anshu wonder why the method works.

Reference: NCERT Class 6, page 113, unnumbered illustration

Why does the sieve work?

Apart from 1, a number is crossed out only as a later multiple of a smaller circled number, which is then a factor other than 1 and itself, so it is composite. A circled number was never crossed out, so no smaller prime divides it; since every number above 1 has a prime factor, no smaller number except 1 divides it either. So it is prime.

The sieve leaves 25 primes up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

Who studied primes long ago?

Eratosthenes, a Greek mathematician who lived around 2200 years ago, developed this method. Primes are the building blocks of all whole numbers, and since the Greek civilisation, more than 2000 years ago, mathematicians have tried to uncover their secrets.

Is there a largest prime, or do the primes go on without end? A mathematician named Euclid found the answer, which comes in a later class. The largest prime anyone has written down would take around 6500 pages to write, so it could be written only on a computer.

What do the Figure it Out questions on primes reveal?

These twelve questions use the list of primes up to 100.

What patterns appear in the list of primes up to 100?

• Q1. No other even number is prime: every even number greater than 2 has 2 as an extra factor.
• Q2. The smallest gap between successive primes is 1, between 2 and 3. The largest gap up to 100 is 8, between 89 and 97.
• Q6. Prime pairs that swap digits, like 13 and 31, are 17 and 71, 37 and 73, and 79 and 97.
• Q7. Seven consecutive composite numbers are 90 to 96, which lie between the primes 89 and 97.
• Q8. Twin primes differ by 2. Besides 3 and 5 and 17 and 19, they are 5 and 7, 11 and 13, 29 and 31, 41 and 43, 59 and 61, and 71 and 73.

For Q3, the rows of the sieve grid are decades, and they do not hold equal numbers of primes. The decade 91 to 100 has the fewest; 1 to 10 and 11 to 20 have the most.

How are the reasoning questions on primes answered?

• Q4. 23 and 37 are prime; 51 = 3 × 17 and 26 = 2 × 13 are composite.
• Q5. Primes below 20 whose sum is a multiple of 5 include 2 and 3, 3 and 7, and 2 and 13. Others are 3 and 17, 7 and 13, 11 and 19, and 13 and 17.
• Q9a. True. A number ending in 4 is even and greater than 2, so 2 is an extra factor.
• Q9b. False. A product of primes has those primes as factors, so it is composite.
• Q9c. False. Every prime has exactly two factors, 1 and itself.
• Q9d. False. The even number 2 is prime.
• Q9e. True. Every prime after 2 is odd, so the next number is even and greater than 2, hence composite.
• Q10. Only 105 = 3 × 5 × 7. The others are 45 = 3 × 3 × 5, 60 = 2 × 2 × 3 × 5, 91 = 7 × 13 and 330 = 2 × 3 × 5 × 11.
• Q11. None. A number made from 2, 4 and 5 ends in 2 or 4, so it is even, or ends in 5, so it is a multiple of 5.
• Q12. Doubling and adding 1 gives a prime for 2 (5), 5 (11), 11 (23), 23 (47), 29 (59) and 41 (83).

What are co-prime numbers, and which treasure pairs are safe?

Jump Jackpot returns with a new rule: a jump size of 1 is not allowed. A pair of treasure numbers is safe for Grumpy if no allowed jump size reaches both. Treasures on 12 and 26 are not safe, as a jump of 2 reaches both; treasures on 4 and 9 are safe.

Definition: Two numbers are co-prime when 1 is their only common factor. So 4 and 9 are co-prime, but 15 and 39 are not.

Safe pairs are exactly the co-prime pairs. Of the five pairs to test, 18 and 35, 15 and 37, and 17 and 69 are co-prime. The pair 30 and 415 shares 5, and 81 and 18 share 3 and 9.

Property: Co-prime pairs have their product as the first common multiple

Anshu noticed that in the idli-vada game the first common multiple is sometimes the product of the two numbers: 15 for 3 and 5, 21 for 3 and 7, 36 for 4 and 9. Each of these pairs is co-prime.

For pairs that share a factor greater than 1, it is smaller than the product: 6 for 3 and 6, 12 for 3 and 12, 30 for 6 and 15.

How does co-prime thread art work?

In thread art, pegs are numbered around a circle and a thread is tied to every peg a fixed step apart, called the thread-gap. With 12 pegs and a thread-gap of 4, the thread visits only pegs 4, 8 and 12.

With 12 pegs, a blue triangle joins pegs 4, 8 and 12. With 13 pegs, a red star passes through all 13 pegs. With 16 pegs, a purple star joins the even-numbered pegs. With 24 pegs, a green square joins pegs 6, 12, 18 and 24.

Reference: NCERT Class 6, page 116, unnumbered illustration

The thread reaches every peg when the number of pegs and the thread-gap are co-prime, as with 13 and 3. When they share a factor, some pegs are missed. The table answers the four picture tasks that follow.

How does prime factorisation break a number into primes?

Where did Anshu go wrong?

Asked whether 56 and 63 are co-prime, Anshu writes 56 = 14 × 4 and 63 = 21 × 3, sees no common factor and says yes. Guna writes 56 = 7 × 8 and 63 = 9 × 7, so 7 is a common factor and they are not co-prime.

One splitting shows some factors, not all of them. Likewise 80 = 16 × 5 and 63 = 9 × 7 share nothing, yet other splittings might. A more systematic approach is needed.

Worked example 5. Write 56 as a product of prime numbers.

Answer: Start with 56 = 4 × 14. Since 14 = 2 × 7, we get 56 = 4 × 2 × 7. Since 4 = 2 × 2, we get 56 = 2 × 2 × 2 × 7. Both 2 and 7 are prime, so the breaking stops.

Definition: Writing a number as a product of primes gives its prime factorisation, and those primes are its prime factors. The prime factors of 56 are 2 and 7.

Any number greater than 1 has a prime factorisation: split composite factors again and again until every factor is prime. The number 1 has none, as no prime divides it. A prime such as 7 is its own prime factorisation.

Property: A number has only one prime factorisation, apart from order

The number 63 comes out as 3 × 3 × 7 or 3 × 7 × 3, with two 3s and one 7 each time. For 36, all four routes give two 2s and two 3s.

From 36, branches start at 2 × 18, 3 × 12, 4 × 9 and 6 × 6. Boxes below break each one further, ending at 2 × 2 × 3 × 3 three times and at 2 × 3 × 2 × 3 once.

Reference: NCERT Class 6, page 118, unnumbered illustration

Does the order of the factors matter?

No. Numbers can be multiplied in any order with the same result, an idea named in a later class as commutativity and associativity of multiplication. Primes are usually written in increasing order, as in 225 = 3 × 3 × 5 × 5 and 30 = 2 × 3 × 5.

Small cubes form a cuboid 3 cubes wide, 2 high and 5 deep, with a red front, yellow top and green side. However it is counted, it shows 30 = 2 × 3 × 5.

Reference: NCERT Class 6, page 119, unnumbered illustration

How is a product of two numbers factorised?

Factorise each part, then put the primes together. For 72 = 12 × 6, the factorisations 12 = 2 × 2 × 3 and 6 = 2 × 3 combine to 72 = 2 × 2 × 2 × 3 × 3. The prime 2 occurs three times: twice from 12 and once from 6.

How are the prime factorisation exercises solved?

A good habit is to divide by the smallest prime that works, repeat, and multiply back at the end to check.

For Q2, one 2, two 3s and one 11 give 2 × 3 × 3 × 11 = 198. For Q5, the smallest number with three different primes is 2 × 3 × 5 = 30, and with four is 2 × 3 × 5 × 7 = 210.

Worked example 6. Find three primes, all less than 30, whose product is 1955 (Q3).

Answer: 1955 ends in 5, so 1955 = 5 × 391. Trying primes on 391 gives 391 = 17 × 23. So 1955 = 5 × 17 × 23.

Worked example 7. Find the prime factorisation of 108 × 75 without multiplying first (Q4b).

Answer: 108 = 2 × 2 × 3 × 3 × 3 and 75 = 3 × 5 × 5. Together, 108 × 75 = 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5.

In the same way, 56 × 25 = 2 × 2 × 2 × 5 × 5 × 7 and 1000 × 81 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5 × 5.

How does prime factorisation test co-primes and divisibility?

For 56 = 2 × 2 × 2 × 7 and 63 = 3 × 3 × 7, the prime 7 appears in both, so they are not co-prime. For 80 = 2 × 2 × 2 × 2 × 5 and 63 = 3 × 3 × 7, no prime is shared.

Could they still share a composite factor? No: its primes would have to appear in both factorisations. So two numbers with no common prime factor are co-prime.

Worked example 8. Are 40 and 231 co-prime? Are 242 and 195 co-prime?

Answer: 40 = 2 × 2 × 2 × 5 and 231 = 3 × 7 × 11 share no prime, so they are co-prime. Also 242 = 2 × 11 × 11 and 195 = 3 × 5 × 13 share no prime, so they are co-prime.

Property: A divisor's prime factorisation sits inside the number's

A number is divisible by another when dividing leaves remainder 0, as 48 ÷ 12 does. When a number is divisible by a second number, every prime of the second, counted as often as it occurs, is included in the first. This checks divisibility without long division.

Worked example 9. Is 168 divisible by 12? Is 75 divisible by 21? Is 42 divisible by 12?

Answer: 168 = 2 × 2 × 2 × 3 × 7 contains 12 = 2 × 2 × 3, and regrouping gives 168 = 12 × 14, so yes. 75 = 3 × 5 × 5 lacks the 7 in 21 = 3 × 7, so no. 42 = 2 × 3 × 7 has one 2, but 12 needs two, so no.

How are the co-prime and divisibility exercises answered?

For Q3, 2 × 3 × 7 = 42 and 3 × 7 × 11 = 231 share 3 and 7, so they are not co-prime, and neither divides the other. For Q4, Guna is right, as the answer key says, for two different primes, since each has only 1 and itself as factors; 3 and 3 are not co-prime.

How can divisibility by 10, 5, 2, 4 and 8 be tested quickly?

Which of 2, 3, 4, …, 10 are factors of 8560? For some of them, a divisibility test answers without long division. Asking whether 8560 is divisible by 10 is the same as asking whether 10 is a factor of it.

What do the last digits show for 10, 5 and 2?

Multiples of 10 end in 0, so 125 is not a multiple of 10. Multiples of 5 end in 0 or 5, so the largest number below 399 divisible by 5 is 395. Multiples of 2 end in 0, 2, 4, 6 or 8, so 682 is divisible by 2.

The multiples of 2 between 399 and 411 are 400, 402, 404, 406, 408 and 410.

Property: Only the last two digits decide divisibility by 4

The last digit alone fails for 4: 12 and 22 both end in 2, yet only 12 is a multiple of 4. Between 330 and 340, the multiples of 4 are 332 and 336; between 1730 and 1740, 1732 and 1736; between 2030 and 2040, 2032 and 2036. The last two digits repeat.

This works because 100 = 4 × 25, so every whole number of hundreds is a multiple of 4. So a number is divisible by 4 exactly when its last two digits form a multiple of 4, and the chapter's three statements are all true. For example, 8536 is divisible by 4, since 36 = 4 × 9.

Why are three digits needed for 8?

Between 120 and 140, the multiples of 8 are 128 and 136; between 1120 and 1140, 1128 and 1136; between 3120 and 3140, 3128 and 3136. As 1000 = 8 × 125, a number is divisible by 8 exactly when its last three digits form a multiple of 8, so the chapter's three statements for 8 all hold.

Two digits are not enough: 116 and 216 both end in 16, but only 216 = 8 × 27 is a multiple of 8. Changing the last two digits of 8560 to make 8552 gives a multiple of 8, since 552 = 8 × 69.

Simple tests for 3, 6, 7 and 9 come in later classes. Dividing directly shows that none of them is a factor of 8560.

How are the divisibility Figure it Out problems solved?

A leap year has 29 days in February, as in 2024. Leap years are multiples of 4, except years divisible by 100 but not by 400. So 2000 was a leap year, but 2100 will not be.

Worked example 10. How many leap years are there from 2024 to 2099 (Q1b)?

Answer: No year from 2024 to 2099 ends in 00, so the leap years are the multiples of 4: 2024, 2028, …, 2096. There are (2096 − 2024) ÷ 4 + 1 = 19 of them.

Q1a depends on the student's birth year. One possible answer: a student born in 2014 and checking in 2026 would list 2016, 2020 and 2024.

• Q2. A four-digit palindrome ends in its first digit, which must be even for divisibility by 4. Testing the last two digits gives 8888 as the largest and 2112 as the smallest. The answer key's 9999 and 1001 are odd, so neither is divisible by 4.
• Q3. Both are sometimes true. Even sums: 2 + 6 = 8 is a multiple of 4, but 2 + 4 = 6 is not. Odd sums: 1 + 3 = 4 is, but 1 + 5 = 6 is not.
• Q5. Guna could check 8 and 5. Divisibility by 8 gives divisibility by 4 and 2, and together with 5 it gives 10. Checking 8 and 10 also works.
• Q6. 5600, 6000 and 77622160 end in 0, and their last three digits (600, 000, 160) are multiples of 8. 572 and 2352 end in 2, so they fail for 5.
• Q7. 10000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5. A number with both 2 and 5 as factors ends in 0, so keep the 2s and 5s apart: 10000 = 16 × 625.

For Q4, the remainder on dividing by 10 is the last digit. Dividing by 5 leaves the last digit, or that digit minus 5 when it is 5 or more. Dividing by 2 leaves 0 for even numbers and 1 for odd ones.

What makes a number special, and how is the prime puzzle solved?

A box holds 9, 16, 25 and 43, and Guna's classmates each pick a number that stands out. Karnawati chooses 9 as the only single-digit number, and Gurupreet chooses it as the only multiple of 3. Murugan chooses 16 as the only even number and only multiple of 4.

Gopika chooses 25 as the only multiple of 5. Yadnyikee chooses 43 as the only prime, and Radha chooses it as the only number that is not a square. Many answers are right, provided each reason is true.

What are the rules of the prime puzzle?

In the prime puzzle, a 3 by 3 grid is filled with primes only. Each row must multiply to the number on its right, and each column to the number below it.

An empty grid has row products 75, 42 and 102 and column products 170, 30 and 63. The solved grid beside it reads 5, 5, 3 in the top row, 2, 3, 7 in the middle row and 17, 2, 3 in the bottom row.

Reference: NCERT Class 6, page 127, unnumbered illustration

Start by factorising every product, then place any prime that can fit only one cell. Each of the four practice puzzles has exactly one solution.

Worked example 11. Solve the puzzle with row products 343, 66 and 44 and column products 28, 154 and 231.

Answer: 343 = 7 × 7 × 7 fills the top row with 7s. Column 1 is 28 = 7 × 2 × 2, so rows 2 and 3 start with 2. Row 2 needs 3 and 11 (66 = 2 × 3 × 11) and row 3 needs 2 and 11 (44 = 2 × 2 × 11). Column 3 is 231 = 7 × 3 × 11, so row 2 ends in 3 and row 3 ends in 11. The grid is 7, 7, 7 / 2, 11, 3 / 2, 2, 11.

Glossary

• Multiple: The result of multiplying a number by 1, 2, 3 and so on; 3, 6 and 9 are multiples of 3.
• Common multiple: A number that is a multiple of each given number; 15, 30 and 45 are common multiples of 3 and 5.
• Factor: A number that divides another exactly, also called a divisor; the factors of 14 are 1, 2, 7 and 14.
• Common factor: A number that is a factor of each given number; 1 and 2 are the common factors of 14 and 36.
• Perfect number: A number whose factors add up to twice the number, such as 6 and 28.
• Prime number: A number with exactly two factors, 1 and itself, such as 2, 3, 5, 7 and 11.
• Composite number: A number with more than two factors, such as 4, 6, 8 and 9; the number 1 is not composite.
• Sieve of Eratosthenes: A method of listing primes by circling each prime in turn and crossing out its later multiples.
• Twin primes: A pair of primes with a difference of 2, such as 3 and 5 or 17 and 19.
• Co-prime numbers: Two numbers whose only common factor is 1, such as 4 and 9.
• Thread-gap: In thread art, the fixed step between the pegs to which the thread is tied.
• Prime factorisation: A number written as a product of primes; for example, 84 = 2 × 2 × 3 × 7.
• Prime factor: A prime that appears in a number's prime factorisation; the prime factors of 56 are 2 and 7.
• Divisibility test: A quick check, without long division, of whether a divisor goes into a number exactly; the tests for 10, 5, 2, 4 and 8 use only the last digits.
• Leap year: A year with 29 days in February: a multiple of 4, except years divisible by 100 but not 400.

Common errors and misconceptions

• Misconception: 1 is a prime number. Correct: 1 has only one factor, so it is neither prime nor composite.
• Misconception: Every even number is composite. Correct: 2 is even and prime; it is the only even prime.
• Misconception: A product of primes can itself be prime. Correct: The product has those primes as extra factors, so it is composite.
• Misconception: If one splitting such as 56 = 14 × 4 and 63 = 21 × 3 shows no common factor, the numbers are co-prime. Correct: One splitting does not show all factors; compare prime factorisations instead.
• Misconception: Different factor trees give different prime factorisations. Correct: Every route gives the same primes the same number of times; only the order can change.
• Misconception: If all the primes of 12 appear in 42, then 42 is divisible by 12. Correct: The counts matter too; 12 has two 2s but 42 has only one.
• Misconception: The last digit shows whether a number is divisible by 4. Correct: 12 and 22 share a last digit, but only 12 is a multiple of 4; use the last two digits.
• Misconception: The first common multiple of two numbers is always their product. Correct: That holds for co-prime pairs; for 4 and 6 it is 12, not 24.

Exam-style questions with model answers

Q1. Find the common factors of 20 and 28. [2 marks]

1. The factors of 20 are 1, 2, 4, 5, 10 and 20, and the factors of 28 are 1, 2, 4, 7, 14 and 28.
2. The numbers in both lists are 1, 2 and 4, so these are the common factors of 20 and 28.

Q2. Without long division, decide whether 8536 is divisible by 4. Give a reason. [2 marks]

1. Only the last two digits decide divisibility by 4, because 100 = 4 × 25, so every whole number of hundreds is a multiple of 4.
2. The last two digits of 8536 form 36, and 36 = 4 × 9. Therefore 8536 is divisible by 4.

Q3. In the idli-vada game, players say “idli” for multiples of 3, “vada” for multiples of 5 and “idli-vada” for common multiples of 3 and 5. The game runs from 1 to 90. How many times is idli said (counting idli-vada), how many times is vada said (counting idli-vada), and how many times is idli-vada said? [3 marks]

1. Idli is said at every multiple of 3 up to 90. Since 90 ÷ 3 = 30, idli is said 30 times, counting the idli-vada turns.
2. Vada is said at every multiple of 5 up to 90. Since 90 ÷ 5 = 18, vada is said 18 times, counting the idli-vada turns.
3. The common multiples of 3 and 5 are the multiples of 15: 15, 30, 45, 60, 75 and 90. So idli-vada is said 6 times.

Q4. Using prime factorisation, decide whether 57 and 85 are co-prime. [3 marks]

1. Factorise the first number: 57 = 3 × 19, so the prime factors of 57 are 3 and 19.
2. Factorise the second number: 85 = 5 × 17, so the prime factors of 85 are 5 and 17.
3. The two lists of prime factors have nothing in common. Numbers with no common prime factor have only 1 as a common factor, so 57 and 85 are co-prime.

Q5. Using prime factorisation, decide whether 168 is divisible by 12 and whether 42 is divisible by 12. [4 marks]

1. Write the factorisations: 168 = 2 × 2 × 2 × 3 × 7 and 12 = 2 × 2 × 3.
2. The factorisation of 12, two 2s and one 3, is included in that of 168. Regrouping gives 168 = (2 × 2 × 3) × (2 × 7) = 12 × 14, so 168 is divisible by 12.
3. Now 42 = 2 × 3 × 7. Both primes of 12, namely 2 and 3, do appear in it.
4. However, 2 occurs twice in 12 but only once in 42, so the factorisation of 12 is not included in that of 42. Therefore 42 is not divisible by 12.

Q6. In Jump Jackpot, Jumpy starts at 0 and uses one jump size throughout. Grumpy places treasures on 28 and 70. Which jump sizes reach both treasures? If a jump size of 1 is not allowed, is this pair safe for Grumpy? [5 marks]

1. A jump size lands on a treasure only when the treasure's number is a multiple of it, that is, when the jump size is a factor of that number.
2. The factors of 28 are 1, 2, 4, 7, 14 and 28. The factors of 70 are 1, 2, 5, 7, 10, 14, 35 and 70.
3. The common factors are 1, 2, 7 and 14, so these four jump sizes reach both treasures.
4. If the jump size 1 is banned, Jumpy can still choose 2, 7 or 14 and collect both treasures.
5. So the pair is not safe. A safe pair must be co-prime, but 28 and 70 share the factors 2, 7 and 14.

Q7. Find the smallest number that is a multiple of every number from 1 to 10. Show the steps. [5 marks]

1. A multiple of 8 is also a multiple of 2 and 4, and a multiple of 9 is also a multiple of 3. So start with 8 and 9.
2. The first common multiple of 8 and 9 is 72. As 72 is a multiple of both 2 and 3, it is also a multiple of 6.
3. The multiples of 72 are 72, 144, 216, 288, 360, … . The first ending in 0 or 5 is 360, so 360 is the smallest multiple of 1, 2, 3, 4, 5, 6, 8, 9 and 10.
4. 360 is not a multiple of 7. Listing 360, 720, 1080, 1440, 1800, 2160, 2520, the first multiple of 7 is 2520 = 7 × 360.
5. So 2520 is the smallest number that is a multiple of every number from 1 to 10.

Q8. Fill a 3 by 3 grid with primes only so that the row products, top to bottom, are 105, 20 and 30, and the column products, left to right, are 28, 125 and 18. [6 marks]

1. Factorise every product. Rows: 105 = 3 × 5 × 7, 20 = 2 × 2 × 5 and 30 = 2 × 3 × 5. Columns: 28 = 2 × 2 × 7, 125 = 5 × 5 × 5 and 18 = 2 × 3 × 3.
2. The middle column multiplies to 125, whose only prime factor is 5, so all three middle cells are 5.
3. The prime 7 appears only in row 1 and column 1, so the top-left cell is 7. Row 1 then ends in 3, since 7 × 5 × 3 = 105.
4. Column 1 needs 2 × 2 below the 7, so rows 2 and 3 both start with 2.
5. Row 2 is 2 × 5 × 2 = 20, so it ends in 2. Row 3 is 2 × 5 × 3 = 30, so it ends in 3.
6. Check column 3: 3 × 2 × 3 = 18. The grid reads 7, 5, 3 / 2, 5, 2 / 2, 5, 3, and every product is correct.

Key takeaways

• A common multiple is a multiple of each given number; in the idli-vada game, players say idli-vada exactly at the common multiples.
• A factor divides a number exactly, and the jump sizes that reach two treasures are the common factors of the two numbers.
• A prime has exactly two factors, 1 and itself; a composite number has more than two; and 1 is neither prime nor composite.
• The Sieve of Eratosthenes circles each prime in turn and crosses out its later multiples, leaving only primes circled.
• Two numbers are co-prime when their only common factor is 1; for such pairs, the first common multiple is their product.
• Every number greater than 1 has exactly one prime factorisation, apart from the order in which the primes are written.
• Numbers are co-prime when their prime factorisations share no prime; a divisor's prime factorisation, with its counts, is included in the number's.
• Divisibility by 10, 5 and 2 depends on the last digit, by 4 on the last two digits, and by 8 on the last three digits.

Test yourself

What is the first number at which players say “idli-vada” when the numbers are 3 and 5?

It is 15, the first number that is a multiple of both 3 and 5.

What are the factors of 24?

They are 1, 2, 3, 4, 6, 8, 12 and 24; each divides 24 exactly, so each jump size reaches 24.

Why is 1 neither prime nor composite?

A prime needs exactly two factors and a composite needs more than two. The number 1 has only one factor.

Name the twin primes between 40 and 75.

They are 41 and 43, 59 and 61, and 71 and 73; each pair differs by 2.

Are 18 and 35 co-prime?

Yes. 18 = 2 × 3 × 3 and 35 = 5 × 7 share no prime factor, so their only common factor is 1.

What is the prime factorisation of 72?

Using 72 = 12 × 6 and combining the primes gives 72 = 2 × 2 × 2 × 3 × 3.

Is 75 divisible by 21?

No. 21 = 3 × 7, but 75 = 3 × 5 × 5 has no 7 in its prime factorisation.

Which digits decide whether a number is divisible by 8?

The last three digits; for example, 8560 is divisible by 8 because 560 = 8 × 70.