Question 18 of 19intermediate🔍 AnalyzeShort Answer3 marks

Find the digital roots of some consecutive multiples of (i) 3, (ii) 4, and (iii) 6. What do you notice?

Correct Answer

(i) Digital roots of multiples of 3: 3, 6, 9, 3, 6, 9, ... ∴ Cycles through 3, 6, 9

(ii) Digital roots of multiples of 4: 4, 8, 3, 7, 2, 6, 1, 5, 9, 4, 8, ... ∴ Cycles through all values

(iii) Digital roots of multiples of 6: 6, 3, 9, 6, 3, 9, ... ∴ Cycles through 6, 3, 9

Observation: The digital root is related to the remainder when divided by 9.

  • Multiples of 3: Since 3 divides 9, pattern cycles through multiples of 3 (3, 6, 9).
  • Multiples of 4: Since 4 and 9 are coprime, pattern cycles through all values 1-9.
  • Multiples of 6: Since 6 and 9 share GCD 3, pattern cycles through multiples of 3.
Exercise: Digital Roots | Q: 3 | (Page Page 19)
For More Understanding

Explanation

The digital root is related to the remainder when divided by 9. For multiples of 3: since 3 divides 9, the pattern cycles through values that are multiples of 3 (3, 6, 9). For multiples of 4: since 4 and 9 are coprime, the pattern cycles through all values 1-9. For multiples of 6: since 6 and 9 share GCD 3, the pattern cycles through values that are multiples of 3.

Solution Steps

  1. Calculate digital roots of consecutive multiples of 3

  2. Calculate digital roots of consecutive multiples of 4

  3. Calculate digital roots of consecutive multiples of 6

  4. Observe the pattern relates to remainder when divided by 9

  5. Identify the relationship between the multiplier and 9

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