Question

Determine if the statement is TRUE or FALSE. Justify your answer. All numbers under discussion are

integers.

10. Given a positive integer N ≥ 10, form the number N' by removing the ones digit from N

and subtracting this digit from the remaining truncated integer. (For example, if N = 1309,

N' = 130 0 9 = 121.) If N' is divisible by 11, then N is divisible by 11.

Answer 1

We can express a positive integer $N\geq 10$ as $N=10A+B$ , where $B$ is the ones digit.

(For example, $N=1309=10\cdot 130+9$ , $A=130$ and $B=9$ )

Then $N^\prime =A-B$ and $N^\prime$ is divisible by $11$ .

$N=10A+B=11A-A+B=11A-(A-B)=11A-N^\prime$

Both $11A$ and $N^\prime$ are divisible by $11.$ Therefore, $N$ is divisible by $11$ .