NCERT Solutions Ganita Prakash (Part 1) Chapter 5 Digital Roots — Figure it Out

Book page 131 Updated on2026-09-05

Q1.

The digital root of an 8-digit number is 5. What will be the digital root of 10 more than that number?

Answer

The new digital root is 6.

Digital root 5 means the number is 5 more than a multiple of 9
10 = 9 + 1, so adding 10 adds 1 to the leftover
New leftover = 5 + 1 = 6

Test it with an actual 8-digit number:

50000000 → digit sum 5 → root 5
50000000 + 10 = 50000010 → digit sum 6 → root 6 ✓

14000000 → digit sum 5 → root 5
14000010 → digit sum 6 → root 6 ✓

Tip: if the starting root had been 8 or 9, the answer would wrap round: 8 + 1 = 9, and 9 + 1 = 10 → 1. Here 5 + 1 = 6 is safely under 9, so no wrap is needed.

Q2.

Write any number. Generate a sequence of numbers by repeatedly adding 11. What would be the digital roots of this sequence of numbers? Share your observations.

Answer

Start at 10 and keep adding 11:

Number	10	21	32	43	54	65	76	87	98	109	120
Digital root	1	3	5	7	9	2	4	6	8	1	3

Observations.

The digital root goes up by 2 at every step, wrapping round after 9.
All nine roots appear, and the whole pattern repeats after 9 terms.
Reading the roots in order gives 1, 3, 5, 7, 9, 2, 4, 6, 8 — the odd roots first, then the even ones.

Why 2: 11 = 9 + 2, so adding 11 adds 2 to the remainder on division by 9. Since 2 and 9 share no factor, repeatedly stepping by 2 lands on every one of the nine roots before returning to the start.

Check it yourself: begin at 7 instead — 7, 18, 29, 40, 51, … The roots are 7, 9, 2, 4, 6, … Same step of 2, different starting point.

Q3.

What will be the digital root of the number 9a + 36b + 13?

Answer

The digital root is 4, whatever whole numbers a and b are.

9a + 36b + 13
= 9a + 36b + 9 + 4
= 9(a + 4b + 1) + 4

The bracket is multiplied by 9, so that part is a multiple of 9 and contributes nothing to the leftover. What remains is 4.

Check with a = 2, b = 1: 18 + 36 + 13 = 67 → 6 + 7 = 13 → 1 + 3 = 4 ✓
Check with a = 0, b = 0: 13 → 1 + 3 = 4 ✓
Check with a = 5, b = 3: 45 + 108 + 13 = 166 → 13 → 4 ✓

Q4.

Make conjectures by examining if there are any patterns or relations between (i) the parity of a number and its digital root. (ii) the digital root of a number and the remainder obtained when the number is divided by 3 or 9.

Answer

(i) Parity and digital root — there is no relation.

Number	Parity	Digital root	Parity of the root
12	even	3	odd
24	even	6	even
15	odd	6	even
27	odd	9	odd

All four combinations occur, so knowing the parity tells you nothing about the digital root and vice versa.

Why: parity is about the remainder on division by 2, the digital root about the remainder on division by 9. Since 2 and 9 share no factor, the two pieces of information are completely independent.

(ii) Digital root and remainders — there is a clean relation both times.

Division by 9: the remainder equals the digital root, except that a digital root of 9 means remainder 0.

Division by 3: the remainder is what the digital root itself leaves on division by 3.

Digital root	Remainder ÷ 9	Remainder ÷ 3
1, 4, 7	1, 4, 7	1
2, 5, 8	2, 5, 8	2
3, 6	3, 6	0
9	0	0

Why: a number and its digit sum leave the same remainder on division by 9, and — because 9 is a multiple of 3 — the same remainder on division by 3 as well. Repeating the digit sum keeps both remainders fixed. So the single digit at the end carries both answers.

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