Class 6 Mathematics · Chapter 5

5. Prime Time

Answers to all questions on factors, multiples, primes, co-primes, prime factorisation and divisibility rules. CBSE / NCERT Class 6 Ganita Prakash.

5.1 Common Multiples and Common Factors

★Find other numbers that are multiples of both 3 and 5. What are these numbers called?

15, 30, 45, 60, 75, 90, … These are called common multiples of 3 and 5.

5.1 — Figure it Out (Idli-Vada)

1At what number is ‘idli-vada’ said for the 10th time?

‘Idli-vada’ is said at 15, 30, 45, … (multiples of 15).

10th time = 10 × 15 = 150

2If the game is played for the numbers 1 to 90, find out how many times the children say (a) ‘idli’ (b) ‘vada’ (c) ‘idli-vada’.

(a) Multiples of 3 up to 90: 90 ÷ 3 = 30 times

(b) Multiples of 5 up to 90: 90 ÷ 5 = 18 times

3What if the game was played till 900? How would your answers change?

‘Idli’: 900 ÷ 3 = 300 times

‘Vada’: 900 ÷ 5 = 180 times

‘Idli-vada’: 900 ÷ 15 = 60 times

Each answer becomes 10 times bigger, because 900 is 10 times 90.

4Is Fig. 5.1 related to the ‘idli-vada’ game? Draw the figure if the game is played till 60.

Yes. The left circle has the ‘idli’ numbers, the right circle has the ‘vada’ numbers, and the common part has the ‘idli-vada’ numbers.

Only multiples of 3	Common multiples of 3 and 5	Only multiples of 5
3, 6, 9, 12, 18, 21, 24, 27, 33, 36, 39, 42, 48, 51, 54, 57	15, 30, 45, 60	5, 10, 20, 25, 35, 40, 50, 55

★Play the game with (a) 2 and 5 (b) 3 and 7 (c) 4 and 6, up to 60.

Pair	Only ‘idli’ numbers	Only ‘vada’ numbers	‘Idli-vada’ (common multiples)
2 and 5	2, 4, 6, 8, 12, 14, 16, 18, 22, 24, … (even numbers not ending in 0)	5, 15, 25, 35, 45, 55	10, 20, 30, 40, 50, 60
3 and 7	3, 6, 9, 12, 15, 18, 24, 27, 30, 33, 36, 39, 45, 48, 51, 54, 57, 60	7, 14, 28, 35, 49, 56	21, 42
4 and 6	4, 8, 16, 20, 28, 32, 40, 44, 52, 56	6, 18, 30, 42, 54	12, 24, 36, 48, 60

★One of the numbers was 4 and nobody said just ‘vada’. Which could be the other number: 2, 3, 5, 8, 10?

Nobody says just ‘vada’ when every multiple of the bigger number is also a multiple of the smaller number.

With 8: every multiple of 8 is a multiple of 4. ✓

With 2: 4 is the bigger number, and every multiple of 4 is a multiple of 2. ✓

So the other number could be 8 or 2.

★What jump sizes can reach both 15 and 30?

Common factors of 15 and 30: 1, 3, 5, 15

★In the table of 31 to 70: What is common among the shaded numbers? The circled numbers? Which are both?

1. All shaded numbers are multiples of 3.

2. All circled numbers are multiples of 4.

3. 36, 48 and 60 are both shaded and circled. They are common multiples of 3 and 4.

5.1 — Figure it Out

1Find all multiples of 40 that lie between 310 and 410.

320, 360, 400

2Who am I? (a) Less than 40, one factor is 7, digit sum is 8. (b) Less than 100, two factors are 3 and 5, one digit is 1 more than the other.

(a) Multiples of 7 less than 40: 7, 14, 21, 28, 35. Only 35 has digit sum 8. Answer: 35

(b) Multiples of 15 less than 100: 15, 30, 45, 60, 75, 90. In 45, 5 is one more than 4. Answer: 45

3Find a perfect number between 1 and 10.

Factors of 6: 1, 2, 3, 6. Sum = 12 = 2 × 6.

So 6 is a perfect number.

4Find the common factors of: (a) 20 and 28 (b) 35 and 50 (c) 4, 8 and 12 (d) 5, 15 and 25

(a) 1, 2, 4

(b) 1, 5

(d) 1, 5

5Find any three numbers that are multiples of 25 but not multiples of 50.

25, 75, 125 (also 175, 225, …)

6Both numbers are smaller than 10. The first ‘idli-vada’ is said after 50. What could the two numbers be?

The first common multiple must be more than 50.

7 and 8 → 56; 7 and 9 → 63; 8 and 9 → 72

So the pair can be 7 and 8, 7 and 9 or 8 and 9.

7Grumpy has kept treasures on 28 and 70. What jump sizes will land on both numbers?

Factors of 28: 1, 2, 4, 7, 14, 28

Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70

Jump sizes (common factors): 1, 2, 7, 14

8Guna erased all numbers except the common multiples 24, 48 and 72. Find the numbers and fill in the regions.

The first common multiple is 24. So the numbers can be 6 and 8.

Only multiples of 6	Common multiples	Only multiples of 8
6, 12, 18, 30, 36, 42, 54, 60, 66	24, 48, 72	8, 16, 32, 40, 56, 64

Other pairs whose first common multiple is 24 also work, for example 3 and 8, or 8 and 12.

9Find the smallest number that is a multiple of all the numbers from 1 to 10, except 7.

It must be a multiple of 8, 9 and 5. 8 × 9 × 5 = 360

10Find the smallest number that is a multiple of all the numbers from 1 to 10.

360 × 7 = 2520

5.2 Prime Numbers

★Find the different ways in which Guna can arrange 12 figs and Anshu can arrange 7 figs in a rectangular manner.

12 figs: 1 × 12, 2 × 6, 3 × 4, 4 × 3, 6 × 2, 12 × 1

7 figs: only 1 × 7 or 7 × 1

★How many prime numbers are there from 21 to 30? How many composite numbers?

Primes: 23 and 29 → 2 primes.

Composite numbers: 21, 22, 24, 25, 26, 27, 28, 30 → 8 composite numbers.

5.2 — Figure it Out

1Is there any other even prime besides 2?

No. Every other even number has 2 as a factor, so it has more than two factors.

2What is the smallest difference between two successive primes up to 100? What is the largest?

Smallest difference = 3 − 2 = 1

Largest difference = 97 − 89 = 8

3Are there an equal number of primes in every row (decade) of the table? Which decades have the least and the most primes?

Decade	1–10	11–20	21–30	31–40	41–50	51–60	61–70	71–80	81–90	91–100
Primes	4	4	2	2	3	2	2	3	2	1

No, the numbers are not equal.

Least: 91–100 (only 97).

Most: 1–10 and 11–20 (4 primes each).

4Which of these numbers are prime: 23, 51, 37, 26?

23 and 37 are prime. 51 = 3 × 17 and 26 = 2 × 13 are composite.

5Write three pairs of prime numbers less than 20 whose sum is a multiple of 5.

2 + 3 = 5

3 + 7 = 10

7 + 13 = 20 (also 3 + 17 = 20, 11 + 19 = 30)

613 and 31 are primes with the same digits. Find such pairs of primes up to 100.

17 and 71 37 and 73 79 and 97

7Find seven consecutive composite numbers between 1 and 100.

90, 91, 92, 93, 94, 95, 96 (91 = 7 × 13)

8Find the twin primes between 1 and 100.

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73)

9Identify whether each statement is true or false. Explain.

(a) There is no prime number whose units digit is 4. — True. Such a number is even and bigger than 2, so it has 2 as a factor.

(b) A product of primes can also be prime. — False. A product of primes has those primes as factors, so it is composite.

(d) All even numbers are composite numbers. — False. 2 is even and prime.

(e) For every prime other than 2, the next number is composite. — True. Every other prime is odd, so the next number is even and bigger than 2.

10Which of these is the product of exactly three distinct primes: 45, 60, 91, 105, 330?

45 = 3 × 3 × 5; 60 = 2 × 2 × 3 × 5; 91 = 7 × 13; 330 = 2 × 3 × 5 × 11

105 = 3 × 5 × 7 is the only one.

11How many three-digit prime numbers can you make using each of 2, 4 and 5 once?

The numbers are 245, 254, 425, 452, 524, 542.

Each ends in 2, 4 or 5, so each is divisible by 2 or 5. So none of them is prime.

123 is prime and 2 × 3 + 1 = 7 is also prime. Find at least five other such primes.

2 → 2 × 2 + 1 = 5

5 → 2 × 5 + 1 = 11

11 → 2 × 11 + 1 = 23

23 → 2 × 23 + 1 = 47

29 → 2 × 29 + 1 = 59

41 → 2 × 41 + 1 = 83

5.3 Co-prime Numbers

★Check if these pairs are safe: (a) 15 and 39 (b) 4 and 15 (c) 18 and 29 (d) 20 and 55

(a) Not safe — 3 is a common factor.

(b) Safe

(d) Not safe — 5 is a common factor.

★Which pairs are co-prime? (a) 18 and 35 (b) 15 and 37 (c) 30 and 415 (d) 17 and 69 (e) 81 and 18

Co-prime: (a) 18 and 35, (b) 15 and 37, (d) 17 and 69.

Not co-prime: (c) 30 and 415 (common factor 5), (e) 81 and 18 (common factor 9).

★Find examples where the first common multiple is equal to, and less than, the product of the two numbers. How is it related to co-primes?

Equal to the product: 3 and 5 → 15; 4 and 9 → 36; 2 and 7 → 14.

Less than the product: 4 and 6 → 12 (product 24); 6 and 9 → 18 (product 54).

When the two numbers are co-prime, the first common multiple is their product.

★Co-prime art: Is touching every peg related to the number of pegs and the thread-gap being co-prime?

Yes. If the number of pegs and the thread-gap are co-prime, the thread touches every peg. Otherwise, some pegs are left out.

Pegs and thread-gap	Co-prime?	Result
15 pegs, gap 10	No (common factor 5)	Thread touches only 3 pegs
10 pegs, gap 7	Yes	Thread touches all 10 pegs
14 pegs, gap 6	No (common factor 2)	Thread touches only 7 pegs
8 pegs, gap 3	Yes	Thread touches all 8 pegs

5.4 Prime Factorisation — Figure it Out

1Find the prime factorisations of: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000.

Number	Prime factorisation
64	2 × 2 × 2 × 2 × 2 × 2
104	2 × 2 × 2 × 13
105	3 × 5 × 7
243	3 × 3 × 3 × 3 × 3
320	2 × 2 × 2 × 2 × 2 × 2 × 5
141	3 × 47
1728	2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
729	3 × 3 × 3 × 3 × 3 × 3
1024	2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
1331	11 × 11 × 11
1000	2 × 2 × 2 × 5 × 5 × 5

2The prime factorisation of a number has one 2, two 3s and one 11. What is the number?

2 × 3 × 3 × 11 = 198

3Find three prime numbers, all less than 30, whose product is 1955.

1955 = 5 × 17 × 23

4Find the prime factorisation without multiplying first: (a) 56 × 25 (b) 108 × 75 (c) 1000 × 81

(a) 56 = 2 × 2 × 2 × 7; 25 = 5 × 5. So 56 × 25 = 2 × 2 × 2 × 5 × 5 × 7

(b) 108 = 2 × 2 × 3 × 3 × 3; 75 = 3 × 5 × 5. So 108 × 75 = 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5

5What is the smallest number whose prime factorisation has (a) three different primes (b) four different primes?

(a) 2 × 3 × 5 = 30

(b) 2 × 3 × 5 × 7 = 210

5.4 — Figure it Out (Co-prime and Divisibility)

1Are these pairs co-prime? (a) 30 and 45 (b) 57 and 85 (c) 121 and 1331 (d) 343 and 216

(a) 30 = 2 × 3 × 5; 45 = 3 × 3 × 5. Common factors 3 and 5 → Not co-prime

(b) 57 = 3 × 19; 85 = 5 × 17. No common prime → Co-prime

(d) 343 = 7 × 7 × 7; 216 = 2 × 2 × 2 × 3 × 3 × 3. No common prime → Co-prime

2Is the first number divisible by the second? (a) 225 and 27 (b) 96 and 24 (c) 343 and 17 (d) 999 and 99

(a) 225 = 3 × 3 × 5 × 5; 27 = 3 × 3 × 3. 225 has only two 3s → No

(b) 96 = 2 × 2 × 2 × 2 × 2 × 3; 24 = 2 × 2 × 2 × 3. Included → Yes

(d) 999 = 3 × 3 × 3 × 37; 99 = 3 × 3 × 11. 11 is missing → No

3First number = 2 × 3 × 7, second number = 3 × 7 × 11. Are they co-prime? Does one divide the other?

They are not co-prime, because 3 and 7 are common factors.

Neither divides the other. The first has 2 (not in the second) and the second has 11 (not in the first).

4Guna says, “Any two prime numbers are co-prime.” Is he right?

Yes, for two different primes. The only factors of a prime are 1 and itself, so two different primes have only 1 as a common factor.

5.5 Divisibility Tests

★Divisibility by 10, 5 and 2.

Multiples of 10 end with 0. 125 does not end with 0, so it is not a multiple of 10. 8560 ends with 0, so it is divisible by 10. Yes, we agree with the statement.

Multiples of 5 end with 0 or 5. The largest number less than 399 divisible by 5 is 395. 8560 is divisible by 5.

Multiples of 2 end with 0, 2, 4, 6 or 8. 682 and 8560 are divisible by 2.

Multiples of 2 between 399 and 411: 400, 402, 404, 406, 408, 410.

★Divisibility by 4.

Between 330 and 340: 332, 336. Between 1730 and 1740: 1732, 1736. Between 2030 and 2040: 2032, 2036.

Observation: the last two digits are 32 and 36 in each case.

8536 is divisible by 4, because 36 is divisible by 4.

All three statements are true. 100 is divisible by 4, so only the number formed by the last two digits decides divisibility by 4.

★Divisibility by 8.

Between 120 and 140: 128, 136. Between 1120 and 1140: 1128, 1136. Between 3120 and 3140: 3128, 3136.

Observation: the last three digits decide divisibility by 8, because 1000 is divisible by 8.

8560 is already a multiple of 8. Changing the last two digits, 8504, 8512, 8552 or 8592 are also multiples of 8.

All three statements are true.

5.5 — Figure it Out

1(a) From the year you were born till now, which years were leap years? (b) From 2024 till 2099, how many leap years are there?

(a) Example: born in 2014 → leap years 2016, 2020, 2024.

(b) 2024, 2028, …, 2096. (2096 − 2024) ÷ 4 + 1 = 19 leap years

2Find the largest and smallest 4-digit numbers that are divisible by 4 and are palindromes.

A 4-digit palindrome ends with its first digit. To be divisible by 4, it must be even, and its last two digits must make a multiple of 4.

Largest: 8888 (88 ÷ 4 = 22). Answer: 8888

Smallest: 2112 (12 ÷ 4 = 3). Answer: 2112

9999 and 1001 are odd numbers, so they cannot be divisible by 4.

3Always, sometimes or never true? (a) Sum of two even numbers gives a multiple of 4. (b) Sum of two odd numbers gives a multiple of 4.

(a) Sometimes true. 2 + 6 = 8 is a multiple of 4, but 2 + 4 = 6 is not.

(b) Sometimes true. 1 + 3 = 4 is a multiple of 4, but 1 + 5 = 6 is not.

4Find the remainders when 78, 99, 173, 572, 980, 1111, 2345 are divided by (a) 10 (b) 5 (c) 2.

Number	÷ 10	÷ 5	÷ 2
78	8	3	0
99	9	4	1
173	3	3	1
572	2	2	0
980	0	0	0
1111	1	1	1
2345	5	0	1

5Guna checked 14560 for divisibility by only two of 2, 4, 5, 8, 10 and declared it divisible by all. Which two?

8 and 5 (or 8 and 10).

A number divisible by 8 is also divisible by 4 and 2. A number divisible by 2 and 5 is divisible by 10.

6Which numbers are divisible by all of 2, 4, 5, 8 and 10: 572, 2352, 5600, 6000, 77622160?

5600, 6000 and 77622160. Each ends in 0, and its last three digits (600, 000, 160) are divisible by 8.

7Write two numbers whose product is 10000. Neither should have 0 as the units digit.

10000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5 = 16 × 625

Answer: 16 × 625

5.6 Fun with Numbers

★Within each box, say how each number is special.

Box	Special reasons (examples)
5, 7, 12, 35	5: smallest number. 7: the only odd number that is not a multiple of 5. 12: the only even number. 35: the only 2-digit odd number (35 = 5 × 7).
3, 8, 11, 24	3: smallest and the only odd single-digit number. 8: the only cube (2 × 2 × 2). 11: the only 2-digit prime. 24: the only 2-digit even number (24 = 3 × 8).
27, 3, 123, 31	27: the only cube. 3: the only single-digit number. 123: the only 3-digit number. 31: the only number that is not a multiple of 3.
17, 27, 44, 65	17: the only prime. 27: the only multiple of 3. 44: the only even number. 65: the only multiple of 5.

★A prime puzzle: Fill the grids with primes so that each row product and column product is correct.

Puzzle 1

7	5	3	105
2	5	2	20
2	5	3	30
28	125	18

Puzzle 2

2	2	2	8
3	5	7	105
5	7	2	70
30	70	28

Puzzle 3

3	7	3	63
3	3	3	27
5	2	19	190
45	42	171

Puzzle 4

7	7	7	343
2	11	3	66
2	2	11	44
28	154	231

The last row of each table shows the column products.

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