EN हिं

NCERT Solutions Ganita Prakash (Part 1) Chapter 5 .2 Checking Divisibility Quickly — In-text Questions

Book page 1235 Updated on2026-09-05

Q1.
Similarly, explain using algebra why the divisibility shortcuts for 5, 2, 4, and 8 work.
Answer

Write the number in general form as a sum of place values:

… + 10000e + 1000d + 100c + 10b + a

Now split it in the right place for each divisor.

Divisibility by 5 and by 2. Every place value above the units — 10, 100, 1000, … — is a multiple of 10, so it is a multiple of both 5 and 2.

N = (a multiple of 10) + a

So N is divisible by 5 only when a is 0 or 5, and by 2 only when a is even. Only the units digit matters.

Divisibility by 4. 100 = 4 × 25, and every higher place value is a multiple of 100. So

N = (a multiple of 100) + 10b + a
= (a multiple of 4) + (the last two digits)

Hence N is divisible by 4 exactly when the two-digit number formed by the last two digits is. For 2856 → 56 = 4 × 14 ✓; for 1586 → 86 = 4 × 21 + 2 ✗.

Divisibility by 8. 1000 = 8 × 125, and every higher place value is a multiple of 1000. So

N = (a multiple of 1000) + 100c + 10b + a
= (a multiple of 8) + (the last three digits)

Hence N is divisible by 8 exactly when the last three digits form a multiple of 8. For 2856 → 856 = 8 × 107 ✓; for 6686 → 686 = 8 × 85 + 6 ✗.

Why the pattern: in each case we cut the number at the first place value that the divisor already divides. Everything to the left is guaranteed to be a multiple of the divisor, so only the small piece on the right needs checking. 10 works for 2 and 5; 100 is needed for 4; 1000 is needed for 8.
Q2.
Can you say, without actually calculating, which of these numbers are divisible by 9: 999, 909, 900, 90, 990?
Answer

All of them.

999 = 9 × 111   909 = 9 × 101
900 = 9 × 100   90 = 9 × 10   990 = 9 × 110
Why it happens: each of these numbers is built only from the digits 9 and 0. Expanding any such number, every term is either 9 × (a place value) or 0 × (a place value) — and each term is a multiple of 9. A sum of multiples of 9 is a multiple of 9.
Was this helpful? Report an error