Figure It Out: Divisibility & Roots

Divisibility by 9 (Page 126)

1. Check Divisibility by 9:

• (i) 123: Sum $1+2+3=6$ (No).
• (ii) 405: Sum $4+0+5=9$ (Yes).
• (iii) 8888: Sum 32 (No).
• (iv) 93547: Sum 28 (No).
• (v) 358095: Sum 30 (No).

2. Smallest multiple of 9 with no odd digits.

• Digits must be even ($0, 2, 4, 6, 8$).
• Sum must be 9? Impossible with only even numbers.
• Sum must be 18.
• Smallest number requires fewest digits, smallest leading digit.
• Try 288: Sum 18. Even digits. Answer: 288.

3. Multiple of 9 closest to 6000.

• $6000 \div 9$: Remainder is sum of digits $6$.
• $6000 - 6 = 5994$. (Distance 6).
• Next multiple: $5994 + 9 = 6003$. (Distance 3).
• Answer: 6003.

4. Multiples of 9 between 4300 and 4400.

• First multiple: Sum of $43xy$ needs to be 9 or 18.
• $4+3=7$. Need 2 more. $4302$.
• Last multiple: $4392$.
• Sequence: $4302, 4311 \dots 4392$.
• Count: $\frac{4392 - 4302}{9} + 1 = 10 + 1 = 11$.

Digital Roots (Page 131)

1. Digital root of 8-digit number is 5.

• Add 10. $+10$ adds $1+0=1$ to the root.
• New root = $5 + 1 = 6$.

2. Sequence adding 11.

• Adding 11 is like adding 2 to the digital root (since $1+1=2$).
• Sequence: $x, x+2, x+4 \dots$ (mod 9).

3. Root of $9a + 36b + 13$.

• $9a$ and $36b$ are multiples of 9. Their digital root contribution is 9 (or 0).
• Only 13 matters. Root of $13 = 1+3 = 4$.
• Answer: 4.

4. Conjectures:

• (i) Parity: A number and its digital root do not always share parity (e.g., 12 is even, root 3 is odd). But $N - \text{Root}(N)$ is always a multiple of 9.
• (ii) Remainder: Digital root is the remainder mod 9.