Why the Divisibility Test for 9 Works: Powers of Ten Are 1 Mod 9

Work each one out before you check it. Every question has a scratch pad and a calculator.

1. Question 1

   Let $s(n)$ be the sum of the digits of $n$. Using $s(n)$ rather than long division, find the remainder of $60\,517$ on division by $9$.

   $r =$

   Check

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2. Question 2

   Without multiplying out, find the remainder of $25 \cdot 10^4 + 13 \cdot 10^2 + 47$ on division by $9$.

   $r =$

   Check

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3. Question 3

   The five-digit number $3d561$ has thousands digit $d$. Find the digit $d$ ($0 \le d \le 9$) for which $9$ divides $3d561$.

   $d =$

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4. Question 4

   Suppose $10^k - 1 = 9m$ for an integer $m$. Which expression equals $10^{k+1} - 1$?

   a

   $9(10m)$

   b

   $9(m + 1)$

   c

   $9(10m + 9)$

   d

   $9(10m + 1)$

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5. Question 5

   Let $n = 4738$. Without computing $n^2$, find the remainder of $n^2$ on division by $9$.

   $r =$

   Check

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