Track / Stage 5 / 259 of 400 #859 of 1964

Problem 859

AIME late

Geometry Difficulty 5.5 Prove it

Will the quotient and remainder change if the dividend and divisor are increased by three times

将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。

Note: The last sentence is a repetition of the instruction and should not be part of the translation. Here is the corrected output:

19. Will the quotient and remainder change if the dividend and divisor are increased by three times

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

19. Let $a=b \cdot c+d$ , where $a$ is the dividend, $b$ is the divisor, $c$ is the quotient, and $d$ is the remainder. Then $3 a=3 b \cdot c+3 d$ , therefore, the quotient will not change, but the remainder will triple?

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.