Remainders of Multiples

Number Theory Level pending

True or False?

For any triplets of integers $(a, b, c)$ satisfying $bc\neq0$ , if $r$ is the remainder of $a$ when divided by $b$ , then $cr$ is the remainder of $ca$ when divided by $cb$ .

True False

1 solution

Muhammad Rasel Parvej
Dec 12, 2016

False is the answer.

$a=bq+r$ , with $0 \leq r < \mid b \mid$ .

Now consider the integers $c<0$ .

Then $ca=(cb)q+(cr)$ , but the inequality $0 \leq cr <\mid cb \mid$ doesn't hold as $c<0 \implies cr<0$ .

For example, when $a=10, b=7$ , $r=3$ . Now, consider $c=-2$ : $cr=-6$ isn't the remainder of $ca=-20$ when divided by $cb=-14$ .

Remainders of Multiples

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