NCERT Solutions Ganita Prakash (Part 1) Chapter 5 Always, Sometimes, or Never — In-text Questions

Book page 120 Updated on2026-09-05

Q1.

4. If a number is divisible by 12, then the number is also divisible by all the factors of 12.

Answer

Always true.

The number is a multiple of 12, say 12m
12m = 2 × 6 × m = 3 × 4 × m = 1 × 12 × m
So 1, 2, 3, 4, 6 and 12 each appear as a factor

In general, if f is a factor of 12 then 12 = f × t for some whole number t, and so

12m = (f × t) × m = f × (tm)

Take 24, which is divisible by 12. Its factors include every factor of 12: 1, 2, 3, 4, 6, 8, 12, 24.

Why it happens (the picture): a multiple of 12 is m rows of 12. A factor of 12 fills one row of 12 exactly — so it fills every row exactly, and therefore fills the whole array with nothing left over.

Tip: in general, if A is divisible by k, then A is divisible by all the factors of k.

Q2.

5. If a number is divisible by 7, then it is also divisible by any multiple of 7.

Answer

Sometimes true. 42 is divisible by 7 and by 14, but not by 28.

42 = 7 × 6, and 14 = 7 × 2 → 42 ÷ 14 = 3 ✓
42 = 7 × 6, and 28 = 7 × 4 → 42 ÷ 28 = 1.5 ✗

The exact condition falls out of the algebra:

7k is divisible by 7m if and only if m is a factor of k
If k = ym, then 7k ÷ 7m = 7ym ÷ 7m = y

For 42, k = 6. The factors of 6 are 1, 2, 3, 6 — so 42 is divisible by 7 × 1 = 7, 7 × 2 = 14, 7 × 3 = 21 and 7 × 6 = 42, but by no other multiple of 7.

Tip: statement 3 and statement 5 look alike but point opposite ways. Going up to multiples of the number always works; going across to other multiples of 7 does not.

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