Figure it Out — Divisibility by 9 — Number Play

1. 12 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 126

   Find, without dividing, whether the following numbers are divisible by 9: (i) 123 (ii) 405 (iii) 8888 (iv) 93547 (v) 358095

   Hint. Add the digits. If the total is a multiple of 9, so is the number.

   The test: a number is divisible by 9 exactly when the sum of its digits is divisible by 9. This works because every power of 10 leaves remainder 1 when divided by 9 (10 = 9+1, 100 = 99+1, 1000 = 999+1, …), so each digit contributes just itself to the remainder.

   (i) 123 → 1 + 2 + 3 = 6. Not a multiple of 9. Not divisible.

   (ii) 405 → 4 + 0 + 5 = 9. A multiple of 9. Divisible. ✓ (Check: 405 ÷ 9 = 45.)

   (iii) 8888 → 8 + 8 + 8 + 8 = 32. Not a multiple of 9 (27 and 36 are the nearest). Not divisible.

   (iv) 93547 → 9 + 3 + 5 + 4 + 7 = 28. Not a multiple of 9. Not divisible.

   (v) 358095 → 3 + 5 + 8 + 0 + 9 + 5 = 30. Not a multiple of 9. Not divisible. (This one is a deliberate trap: it contains a 9 and ends in 5, and 30 is a multiple of 3, so the number is divisible by 3 but not by 9.)

   Shortcut for long numbers: cast out the 9s as you add. In (v), the 9 and the pair 3+5+... can be ignored or paired to nines, leaving the same conclusion faster.

   ✦ Only (ii) 405 is divisible by 9; the digit sums are 6, 9, 32, 28 and 30 respectively.

2. 23 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 126

   Find the smallest multiple of 9 with no odd digits.

   Hint. The digits available are 0, 2, 4, 6, 8 — all even. What does that force the digit sum to be?

   Step 1 — List what digits are allowed. Only even digits: 0, 2, 4, 6, 8.

   Step 2 — Work out what the digit sum must be. The digit sum has to be a multiple of 9. But a sum of even numbers is always even, so the digit sum cannot be 9 itself — 9 is odd. The next multiple of 9 is 18, which is even, so the digit sum must be at least 18.

   Step 3 — Rule out two-digit numbers. The largest possible two-digit sum using even digits is 8 + 8 = 16, which is short of 18. So no two-digit number works, and we need at least three digits.

   Step 4 — Find the smallest three-digit number with even digits summing to 18. To make a number small, make the leading digit as small as possible. • Leading digit 2: the other two must sum to 16, and the only even pair is 8 + 8 → 288 • Leading digit 4: the other two must sum to 14, e.g. 6 + 8 → 468 • Leading digit 6 or 8 gives even larger numbers.

   So the smallest is 288.

   Check: digits 2, 8, 8 are all even ✓, sum = 18 which is a multiple of 9 ✓, and 288 ÷ 9 = 32 ✓

   ✦ 288 — the digit sum must be even and a multiple of 9, so at least 18, which needs three digits, and 288 is the smallest arrangement.

3. 32 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 126

   Find the multiple of 9 that is closest to the number 6000.

   Hint. Divide 6000 by 9 and look at the multiples on either side.

   Step 1 — Locate 6000 between two multiples of 9. 6000 ÷ 9 = 666.67 (not a whole number, so 6000 itself is not a multiple of 9). The multiple just below: 9 × 666 = 5994 The multiple just above: 9 × 667 = 6003

   Step 2 — Measure both distances. 6000 − 5994 = 6 6003 − 6000 = 3

   Since 3 < 6, the multiple above is nearer.

   A faster route using the digit-sum test. The digit sum of 6000 is 6, so 6000 leaves remainder 6 on division by 9. That immediately tells you the previous multiple is 6 below (5994) and the next is 9 − 6 = 3 above (6003) — and 3 beats 6 without any division at all.

   Check: 6003 → 6 + 0 + 0 + 3 = 9 ✓, and 6003 ÷ 9 = 667 ✓

   ✦ 6003, which is only 3 away, whereas the nearest multiple below (5994) is 6 away.

4. 42 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 126

   How many multiples of 9 are there between the numbers 4300 and 4400?

   Hint. Find the first multiple above 4300 and the last one below 4400, then count the steps between them.

   Step 1 — Find the first multiple of 9 above 4300. 4300 ÷ 9 = 477.8, so the next whole multiplier is 478: 9 × 478 = 4302.

   Step 2 — Find the last multiple of 9 below 4400. 4400 ÷ 9 = 488.9, so the largest whole multiplier is 488: 9 × 488 = 4392.

   Step 3 — Count them. The multipliers run from 478 to 488 inclusive. Counting inclusively: 488 − 478 + 1 = 11

   Why the '+1' matters. Subtracting alone counts the gaps between multiples, not the multiples themselves. There are 10 gaps but 11 numbers — the classic fence-post situation.

   Check by listing: 4302, 4311, 4320, 4329, 4338, 4347, 4356, 4365, 4374, 4383, 4392 — that is 11 numbers ✓

   Sanity check on the size: the interval spans 100, and multiples of 9 occur every 9 numbers, so roughly 100 ÷ 9 ≈ 11 of them. That matches.

   ✦ There are 11 multiples of 9 between 4300 and 4400, running from 4302 to 4392.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 8 Mathematics textbook Ganita Prakash Part 1, Reprint 2026-27 (hegp105.pdf). The chapter is about divisibility reasoning, digital roots and cryptarithms — questions sit in four 'Figure it Out' blocks (pages 122, 126, 131 and 132-135) plus several in-text drills. Every numeric answer here was independently recomputed before being compared with the book's printed answer key: the full 10-row divisibility table was re-derived rule by rule, every cryptarithm was re-solved from scratch, and the divisibility-by-44 and multiple-of-18 digit pairs were found by exhaustive search over the digits.. Questions are referenced from the NCERT textbook for identification.