NCERT Solutions for Class 7th Maths Chapter 3 In-text Questions — Primes and Prime Factorisation

Book page 49 Updated on2026-09-19

Q1.

So far, we have been listing all the factors to find the HCF. This can become cumbersome for numbers with many factors, as you would have observed for the numbers 30 and 50, and 28 and 42. Sometimes, we may also miss some factors which can lead to errors. Can this process be simplified? Can it be made more reliable?

Answer

Yes — prime factorisation simplifies it and makes it reliable.

Listing factors has two weaknesses. It is long for numbers such as 360 or 840, and it is easy to miss one.

Listing method: 30 → 1, 2, 3, 5, 6, 10, 15, 30
50 → 1, 2, 5, 10, 25, 50
Compare the two lists, pick the largest match = 10

Prime method: 30 = 2 × 3 × 5, 50 = 2 × 5 × 5
Common primes: one 2 and one 5
HCF = 2 × 5 = 10

Why it happens: a number has many factors but only one prime factorisation. Working with the primes means working with a short, fixed list instead of a long one you must build yourself — so there is nothing to miss.

Q2.

The number 90 could also have been factorised as 3 × 30 or 2 × 45 or in a few other different ways. Will these all lead to the same prime factors?

Answer

Yes. The prime factors always come out the same, only the order may differ.

90 = 9 × 10 = (3 × 3) × (2 × 5) = 2 × 3 × 3 × 5
90 = 3 × 30 = 3 × (2 × 15) = 3 × 2 × 3 × 5
90 = 2 × 45 = 2 × (9 × 5) = 2 × 3 × 3 × 5
90 = 6 × 15 = (2 × 3) × (3 × 5) = 2 × 3 × 3 × 5

Every route ends with one 2, two 3s and one 5.

Why it happens: however you split the number, you keep breaking composite pieces until nothing composite is left. The pieces you finish with are the primes hidden inside 90 from the start — splitting differently only changes the order in which you dig them out, not which ones are there.

Did you know? This fact — that every number has exactly one prime factorisation apart from order — is so important that it is called the Fundamental Theorem of Arithmetic.

Q3.

Procedure for Prime Factorisation. Can you see what is happening below?

The two figures from the book: a circled number with another number written to its left, a circled number below it in the same way, and a plain number at the foot.

Answer

At each step the number is divided by a prime, and the quotient is written below.

105 ÷ 3 = 35, so write 3 outside and 35 below
35 ÷ 5 = 7, so write 5 outside and 7 below
7 is prime — stop

30 ÷ 2 = 15, so write 2 outside and 15 below
15 ÷ 3 = 5, so write 3 outside and 5 below
5 is prime — stop

Read the numbers outside from top to bottom, then the last number at the bottom:

105 = 3 × 5 × 7
30 = 2 × 3 × 5

Why it happens: each circled number is the product of the number to its left and the number below it — 105 = 3 × 35 and 35 = 5 × 7. Putting these together, 105 = 3 × (5 × 7). Every step peels off one prime, so when the peeling stops, the primes collected are the prime factorisation.

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