NCERT Solutions Ganita Prakash Chapter 3 Figure it Out — Digit sums of numbers

Book page 60 Updated on2026-09-05

Q1.

Digit sum 14. a. Write other numbers whose digits add up to 14. b. What is the smallest number whose digit sum is 14? c. What is the largest 5-digit number whose digit sum is 14? d. How big a number can you form having the digit sum of 14? Can you make an even bigger number?

Answer

a. Any number whose digits add up to 14 will do:

59 → 5 + 9 = 14    77 → 7 + 7 = 14    86 → 8 + 6 = 14    95 → 9 + 5 = 14
158 → 1 + 5 + 8 = 14    356 → 3 + 5 + 6 = 14    806 → 8 + 0 + 6 = 14
1247 → 1 + 2 + 4 + 7 = 14    3407 → 3 + 4 + 0 + 7 = 14    22226 → 2+2+2+2+6 = 14

b. The smallest number is 59.

Why: a 1-digit number can have a digit sum of at most 9, so we need at least 2 digits. To make the number small, keep the tens digit as small as possible: 5 in the tens place leaves 9 for the ones place, and 5 + 9 = 14. A tens digit of 4 would need 10 in the ones place, which is impossible.

c. The largest 5-digit number is 95,000.

Put the biggest digit first: 9
14 − 9 = 5, so the next digit is 5
Nothing is left, so fill the rest with zeros → 95,000
Check: 9 + 5 + 0 + 0 + 0 = 14 ✓

d. There is no biggest such number — you can always make a bigger one.

95 → 9005 → 900005 → 9000000005 → 9000000000000005 …

Each time, push the 5 further to the right and fill the gap with zeros. Zeros add nothing to the digit sum, so the sum stays 14 while the number keeps growing. Hence you can always make an even bigger number.

Q2.

Find out the digit sums of all the numbers from 40 to 70. Share your observations with the class.

Answer

Number	40	41	42	43	44	45	46	47	48	49
Digit sum	4	5	6	7	8	9	10	11	12	13
Number	50	51	52	53	54	55	56	57	58	59
Digit sum	5	6	7	8	9	10	11	12	13	14
Number	60	61	62	63	64	65	66	67	68	69
Digit sum	6	7	8	9	10	11	12	13	14	15

And 70 → 7 + 0 = 7.

Observations

Inside one decade (40–49, 50–59, 60–69) the digit sum goes up by 1 each time.
When we cross into a new decade the digit sum drops by 8: 49 → 13 but 50 → 5, and 59 → 14 but 60 → 6.
The largest digit sum in this range is 15 (at 69) and the smallest is 4 (at 40).
Some digit sums repeat — 5, 6, 7, 8, 9 appear in more than one decade. For example 41, 50 both have digit sum 5.

Why the drop is 8: going from 49 to 50 the ones digit falls from 9 to 0 (−9) while the tens digit rises from 4 to 5 (+1). Altogether the digit sum changes by −9 + 1 = −8.

Q3.

Calculate the digit sums of 3-digit numbers whose digits are consecutive (for example, 345). Do you see a pattern? Will this pattern continue?

Answer

Number	Digit sum	Value	3 × middle digit
123	1 + 2 + 3	6	3 × 2 = 6
234	2 + 3 + 4	9	3 × 3 = 9
345	3 + 4 + 5	12	3 × 4 = 12
456	4 + 5 + 6	15	3 × 5 = 15
567	5 + 6 + 7	18	3 × 6 = 18
678	6 + 7 + 8	21	3 × 7 = 21
789	7 + 8 + 9	24	3 × 8 = 24

The pattern: every digit sum is a multiple of 3, and the sums go up in steps of 3 — 6, 9, 12, 15, 18, 21, 24.

Why it happens: if the middle digit is m, the three consecutive digits are (m − 1), m, (m + 1). Adding them:

(m − 1) + m + (m + 1) = 3 × m

The −1 and the +1 cancel out, so the digit sum is always exactly three times the middle digit — and therefore always a multiple of 3.

Will the pattern continue? No — it has to stop. The next number would need the digits 8, 9, 10, but 10 is not a single digit. So 789 is the last one, and there are only these seven such numbers.

Try This: the same seven sums appear if you write the digits the other way round — 321, 432, 543 … Check that 987 also has digit sum 24.

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