Track / Stage 5 / 204 of 400 #804 of 1964

Problem 804

AIME late

Geometry Difficulty 5.5 Prove it

19. 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 and solution reproduced as published; topic, difficulty and ordering added by this site.