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NCERT Solutions Ganita Prakash (Part 1) Chapter 1 Factors and partner factors — In-text Questions

Book page 2 Updated on2026-09-05

Q1.
Does every number have an even number of factors?
Answer

No. Most numbers do, but the squares do not.

6 : 1 × 6, 2 × 3 → 4 factors (even)
10 : 1 × 10, 2 × 5 → 4 factors (even)
9 : 1 × 9, 3 × 3 → 3 factors (odd)
Why it happens: Writing a number as a product of two factors always pairs a factor with a partner. As long as the two members of every pair are different, the factors can be matched up two by two and the total is even. A number breaks that pattern only when one of its pairs has both members equal — and a pair d × d means the number is d2.

So: every number except a perfect square has an even number of factors.

Q2.
Can you use this insight to find more numbers with an odd number of factors?
Answer

Yes — take any number times itself.

1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25, …

Their factor counts are 1, 3, 3, 5, 3, … — all odd.

NumberFactorsHow many
111
41, 2, 43
91, 3, 93
161, 2, 4, 8, 165
251, 5, 253
361, 2, 3, 4, 6, 9, 12, 18, 369

Every one of these is a square, and no number that is not a square appears on such a list.

Q3.
For instance, 36 has a factor pair 6 × 6 where both numbers are 6. Does this number have an odd number of factors? If every factor of 36 other than 6 has a different factor as its partner, then we can be sure that 36 has an odd number of factors. Check if this is true.
Answer

Yes, 36 has 9 factors — an odd number. Here is the check, pair by pair.

1 × 36 → partners 1 and 36, different
2 × 18 → partners 2 and 18, different
3 × 12 → partners 3 and 12, different
4 × 9 → partners 4 and 9, different
6 × 6 → partner of 6 is 6 itself

Four pairs of distinct factors give 4 × 2 = 8 factors, and the lone 6 adds one more:

8 + 1 = 9 factors — 1, 2, 3, 4, 6, 9, 12, 18, 36
Why it happens: The claim in the question is exactly the general argument. All the factors except the self-partner can be swept into disjoint pairs, contributing an even count; the self-partner is left over on its own, tipping the total to odd. Only a square has such a self-partner, so only a square can have an odd number of factors.
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