NCERT Solutions Ganita Prakash (Part 1) Chapter 1 Square Roots by prime factorisation — In-text Questions

Book page 9 Updated on2026-09-05

Q1.

We know that a perfect square is obtained by multiplying an integer by itself. Will looking at a number's prime factorisation help in determining whether it is a perfect square?

Answer

Yes. If the prime factors can be divided into two equal groups, the product of one group is the square root.

36 = 2 × 2 × 3 × 3 = (2 × 3) × (2 × 3) = 6² → √36 = 6
50 = 2 × 5 × 5 → the 2 has no partner → not a perfect square

Why it works: Squaring m repeats every prime of m exactly twice as often. Reading that backwards, a number is a perfect square precisely when each of its primes appears an even number of times, so the whole factorisation can be dealt out into two identical halves. Halving each exponent gives the root at once.

Q2.

Is 324 a perfect square?

Answer

Yes, 324 = 18².

324 = 2 × 2 × 3 × 3 × 3 × 3
Two equal groups: (2 × 3 × 3) × (2 × 3 × 3)
= (2 × 3 × 3)² = 18²
Or in pairs: (2 × 2) × (3 × 3) × (3 × 3)
Therefore √324 = 18

Why both ways agree: Pairing the primes shows each prime occurs an even number of times — 2 twice, 3 four times. Taking one member from each pair, 2 × 3 × 3 = 18, rebuilds the root. Grouping into two halves and pairing are two views of the same fact, that every exponent is even.

Q3.

Is 156 a perfect square?

Answer

No.

156 = 2 × 2 × 3 × 13
The pair (2 × 2) is fine, but 3 and 13 are left unpaired
So 156 cannot be split into two identical groups

Therefore 156 is not a perfect square.

Check it yourself: 12² = 144 and 13² = 169, so 156 sits between two consecutive squares and cannot be one itself.

Q4.

Find whether 1156 and 2800 are perfect squares using prime factorisation.

Answer

1156 is a perfect square; 2800 is not.

1156 = 2 × 578 = 2 × 2 × 289 = 2 × 2 × 17 × 17
= (2 × 17) × (2 × 17) = 34²
√1156 = 34

2800 = 2 × 1400 = 2 × 2 × 700 = 2 × 2 × 2 × 350 = 2 × 2 × 2 × 2 × 175
= 2 × 2 × 2 × 2 × 5 × 5 × 7
The four 2s pair up and the two 5s pair up, but the single 7 has no partner

So 2800 is not a perfect square.

Why the leftover 7 decides it: In a perfect square every prime must occur an even number of times. In 2800 the prime 7 occurs once, an odd number of times, so no matter how the factors are arranged one 7 will always be stranded.

Tip: 2800 = 400 × 7 = 20² × 7. Multiplying 2800 by 7 would give 19600 = 140².

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