Figure it Out — Digital Roots — Number Play

1. 12 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 131

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

   Hint. Use the fact that the digital root is the remainder mod 9, rather than trying to track the digits.

   Step 1 — Translate the given information. The digital root of a number equals its remainder on division by 9. So if the digital root of n is 5, then n ≡ 5 (mod 9).

   Step 2 — See what adding 10 does. 10 leaves remainder 1 on division by 9, since 10 = 9 + 1. So n + 10 ≡ 5 + 1 = 6 (mod 9).

   The digital root of n + 10 is therefore 6.

   Step 3 — Why the mod-9 route is the right one. It is tempting to say "adding 10 raises the tens digit by 1, so the digit sum goes up by 1". That is true only when the tens digit is not 9 — otherwise carrying changes several digits at once. Working with the remainder avoids the issue completely, because the remainder argument holds no matter how much carrying occurs.

   Example with carrying: take n = 40000091, whose digit sum is 14 → 5 ✓. Then n + 10 = 40000101, digit sum 6 ✓ — still 6, even though three digits changed. Example without: n = 40000001 has digital root 5, and n + 10 = 40000011 has digital root 6 ✓

   The fact that the number has 8 digits is a red herring — the answer does not depend on the length at all.

   ✦ 6, since the digital root is the remainder mod 9 and 10 ≡ 1 (mod 9), so the root increases by exactly 1 regardless of any carrying.

2. 23 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 131

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

   Hint. What remainder does 11 leave on division by 9?

   Step 1 — Take an example. Start at 10 and keep adding 11: 10, 21, 32, 43, 54, 65, 76, 87, 98, 109, 120, …

   Step 2 — Take the digital roots. 10 → 1 21 → 3 32 → 5 43 → 7 54 → 9 65 → 11 → 2 76 → 13 → 4 87 → 15 → 6 98 → 17 → 8 109 → 10 → 1 120 → 3

   So the roots run 1, 3, 5, 7, 9, 2, 4, 6, 8, then back to 1 — a cycle of length 9.

   Step 3 — Explain the pattern. Since 11 = 9 + 2, adding 11 has the same effect on the remainder mod 9 as adding 2. So each digital root is 2 more than the previous one, wrapping round past 9. Starting from 1: 1, 3, 5, 7, 9, then 11 wraps to 2, then 4, 6, 8, then 10 wraps to 1 — the cycle closes after 9 terms.

   Observations worth recording: • The cycle length is 9, and every digit from 1 to 9 appears exactly once in it. This happens because 2 and 9 share no common factor, so repeated steps of 2 reach every residue before returning. • The odd roots come first, then the even ones — a consequence of stepping by 2 through nine positions. • The starting number only shifts where the cycle begins; the pattern of jumps by 2 is the same whatever you start with.

   ✦ The digital roots increase by 2 each time, wrapping past 9 — giving the repeating cycle 1, 3, 5, 7, 9, 2, 4, 6, 8 (all nine roots), because 11 ≡ 2 (mod 9).

3. 32 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 131

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

   Hint. Split off everything that is already a multiple of 9.

   Step 1 — Separate out the multiples of 9. 9a is a multiple of 9. 36b is a multiple of 9, since 36 = 9 × 4. 13 is not, but it can be split as 9 + 4.

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

   Step 2 — Read off the remainder. The first part, 9(a + 4b + 1), is a multiple of 9 and so contributes nothing to the remainder. What is left over is 4.

   Step 3 — Convert to a digital root. Since the digital root equals the remainder mod 9 (using 9 in place of 0), and the remainder here is 4, the digital root is 4.

   Check with numbers: take a = 2, b = 1. Then 9(2) + 36(1) + 13 = 18 + 36 + 13 = 67, and 6 + 7 = 13 → 1 + 3 = 4 ✓ Another: a = 0, b = 0 gives 13 → 1 + 3 = 4 ✓

   Notice the answer does not depend on a or b at all, because every term involving them is already a multiple of 9.

   ✦ 4, since 9a + 36b + 13 = 9(a + 4b + 1) + 4, so the expression always leaves remainder 4 on division by 9 whatever a and b are.

4. 43 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 131

   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.

   Hint. Parity is about division by 2; digital roots are about division by 9. Do 2 and 9 interact?

   (i) Parity and digital root — no consistent relation.

   Test both directions: • Even numbers 4, 6, 8 have digital roots 4, 6, 8 — even. But even numbers 14, 16 have digital roots 5 and 7 — odd. • Odd numbers 5, 7 have odd roots, but odd numbers 15, 27 have roots 6 and 9, one even and one odd.

   So an even number can have either an even or an odd digital root, and likewise for odd numbers. No pattern exists.

   The reason is structural: parity is about the remainder on division by 2, while the digital root is about the remainder on division by 9. Since 2 and 9 share no common factor, knowing one remainder tells you nothing about the other — every combination of the two occurs.

   (ii) Digital root and the remainders mod 3 and mod 9 — a tight relation.

   Division by 9. The digital root is the remainder, with one relabelling: a digital root of 9 corresponds to remainder 0, and any digital root below 9 is the remainder itself.

   Division by 3. Because 3 divides 9, the remainder mod 3 can be read straight off the digital root:

   Digital root	Remainder on division by 3
   1, 4, 7	1
   2, 5, 8	2
   3, 6, 9	0

   Each column groups the roots that differ by 3, which is exactly why they share a remainder mod 3.

   Check: 25 has digital root 7, and 25 ÷ 3 leaves 1 ✓. 38 has digital root 2, and 38 ÷ 3 leaves 2 ✓. 42 has digital root 6, and 42 ÷ 3 leaves 0 ✓

   ✦ (i) No relation — parity concerns division by 2 while the digital root concerns division by 9, and the two share no common factor. (ii) The digital root is the remainder mod 9 (with 9 standing for 0); and mod 3, roots 1/4/7 give remainder 1, roots 2/5/8 give 2, and roots 3/6/9 give 0.

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.