Find the smallest number who gives remainders $75,60,105$ when divided by $90,75,120$

Question

Which is the least number when divided by $90,75,120$ leaves remainder $75,60,105$ respectively? How should approach this question

Answer 1

The Chinese Remainder Theorem gives a systematic way to solve such problems.

For this particular problem, there is a shortcut. We want

$ x \equiv \ 75 \bmod 90 \\ x \equiv \ 60 \bmod 75 \\ x \equiv 105 \bmod 120 $

or, equivalently,

$ x+15 \equiv 0 \bmod 90 \\ x+15 \equiv 0 \bmod 75 \\ x+15 \equiv 0 \bmod 120 $

Therefore, $x+15$ is a common multiple of $90,75,120$ and so is a multiple of $lcm(90,75,120)=1800$.