NCERT Solutions Ganita Prakash Chapter 5 Using prime factorisation — Figure it Out

Book page 122 Updated on2026-09-05

Q1.

Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. a. 30 and 45 b. 57 and 85 c. 121 and 1331 d. 343 and 216

Answer

Pair	Prime factorisation	Common prime factor	Co-prime?
a. 30 and 45	30 = 2 × 3 × 5 45 = 3 × 3 × 5	3 and 5	No
b. 57 and 85	57 = 3 × 19 85 = 5 × 17	none	Yes ✔
c. 121 and 1331	121 = 11 × 11 1331 = 11 × 11 × 11	11	No
d. 343 and 216	343 = 7 × 7 × 7 216 = 2 × 2 × 2 × 3 × 3 × 3	none	Yes ✔

Checks: 3 × 19 = 57 ✔ 5 × 17 = 85 ✔ 7 × 7 × 7 = 343 ✔ 6 × 6 × 6 = 216 ✔

A good guess first: 30 and 45 are both in the 5 table, so they cannot be co-prime. 121 and 1331 are both powers of 11. And 343 = 7³ is odd while 216 = 6³ is even — a promising sign that they share nothing.

Q2.

Is the first number divisible by the second? Use prime factorisation. a. 225 and 27 b. 96 and 24 c. 343 and 17 d. 999 and 99

Answer

The first number is divisible by the second only if the whole prime factorisation of the second sits inside that of the first.

Part	Prime factorisations	Is the second one included?	Divisible?
a. 225 ÷ 27	225 = 3 × 3 × 5 × 5 27 = 3 × 3 × 3	27 needs three 3s, 225 has only two	No
b. 96 ÷ 24	96 = 2 × 2 × 2 × 2 × 2 × 3 24 = 2 × 2 × 2 × 3	yes — 96 = (2 × 2 × 2 × 3) × 2 × 2	Yes ✔ (96 ÷ 24 = 4)
c. 343 ÷ 17	343 = 7 × 7 × 7 17 = 17	17 does not appear in 343 at all	No
d. 999 ÷ 99	999 = 3 × 3 × 3 × 37 99 = 3 × 3 × 11	11 does not appear in 999	No

Long-division check: 225 ÷ 27 = 8 remainder 9 ✘ 96 ÷ 24 = 4 remainder 0 ✔
343 ÷ 17 = 20 remainder 3 ✘ 999 ÷ 99 = 10 remainder 9 ✘

Watch part (a) closely: both 225 and 27 are made only of 3s and 5s, so it looks promising. But how many times a prime occurs matters. 27 wants three 3s and 225 can supply only two — so the division fails.

Q3.

The first number has prime factorisation 2 × 3 × 7 and the second number has prime factorisation 3 × 7 × 11. Are they co-prime? Does one of them divide the other?

Answer

First number = 2 × 3 × 7 = 42
Second number = 3 × 7 × 11 = 231

Are they co-prime? No. They share the primes 3 and 7, so 3, 7 and 21 are all common factors.

Does one divide the other? No, neither divides the other.

Does 42 divide 231? 42 needs a 2, but 231 = 3 × 7 × 11 has no 2 → No
Does 231 divide 42? 231 needs an 11, but 42 = 2 × 3 × 7 has no 11 → No
(Also 231 is bigger than 42, so it could never divide it.)

Check: 231 ÷ 42 = 5 remainder 21 ✘

Remember the two different tests: for co-prime we ask “is any prime shared?”; for divisibility we ask “is the whole factorisation contained?” Two numbers can share some primes and still not divide each other — exactly as here.

Q4.

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

Answer

Yes, Guna is right — as long as the two primes are different.

Factors of a prime p → 1 and p only
Factors of a different prime q → 1 and q only
Since p ≠ q, the only factor they share is 1 → co-prime ✔

Examples:

2 and 3 → common factor only 1 ✔
3 and 11 → common factor only 1 ✔
17 and 89 → common factor only 1 ✔

The one exception to keep in mind: if the two primes are the same number, say 5 and 5, then 5 is a common factor and they are not co-prime. So the correct statement is: any two different prime numbers are co-prime.

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