Question

The number $N$ is a two-digit number with the following properties:

When $N$ is divided by $9,$ the remainder is $1.$
When $N$ is divided by $10,$ the remainder is $3.$

What is the remainder when $N$ is divided by $11?$

$0$

$2$

$4$

$5$

$7$

Answer: E

Concepts:modular arithmetic systematic listing

Difficulty rating: 940

Small Hint:

The remainder when divided by $10$ tells you the units digit of $N$ .

Big Hint:

List the two-digit numbers ending in $3$ and test the condition modulo $9$ .

Video solution:

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Written solution:

The second condition says that $N$ ends in $3$ , so the possibilities are $13,$ $23,$ $33,$ $43,$ $53,$ $63,$ $73,$ $83,$ $93.$ Of these, only $73$ leaves remainder $1$ when divided by $9$ , because its digit sum is $10$ . Finally, $73=6\cdot11+7$ , so the requested remainder is $7$ .

Thus, E is the correct answer.

2016 AMC 8 Problem 5