NCERT Solutions – Class 8 Maths (Ganita Prakash)
Chapter: Number Play
Exercise 5.4
1. The digital root of an 8-digit number is 5. What will be the digital root of 10 more than that number?
Digital root follows modulo 9 pattern.
Given: Digital root = 5 Add 10 → same as adding 1 (since 10 ≡ 1 mod 9)
New digital root = 5 + 1 = 6
Answer: 6
2. Generate a sequence by repeatedly adding 11. What are the digital roots?
Let starting number = a
Sequence: a, a+11, a+22, a+33, …
Since 11 ≡ 2 (mod 9), digital root increases by 2 each time.
Example (start with 1):
1 → 12 → 23 → 34 → 45 → … Digital roots: 1, 3, 5, 7, 9, 2, 4, 6, 8, 1…
Observation: Digital roots form a cyclic pattern increasing by 2 (mod 9).
3. Digital root of 9a + 36b + 13
9a ≡ 0 (mod 9)
36b ≡ 0 (mod 9)
13 ≡ 4 (mod 9)
Total ≡ 4 (mod 9)
Answer: 4
4. Conjectures:
(i) Relation between parity and digital root:
No fixed relation.
Example:
- Even number 14 → digital root = 5 (odd)
- Odd number 13 → digital root = 4 (even)
Conclusion: No direct relation between parity and digital root.
(ii) Relation between digital root and remainder when divided by 3 or 9:
Digital root is directly related to modulo 9.
- If digital root = 9 → number divisible by 9
- Digital root gives remainder when divided by 9
- For division by 3, digital root also helps (since 9 is multiple of 3)
Conclusion: Digital root determines divisibility and remainder for 3 and 9.