Question

A layout designer wants to put an equal number of math problems on each page of a new math manual. When she tries to put 5 problems on each page, she discovers that the last page would only have 4 problems on it. When she tries to put 6 problems on each page, the last page would have only 5 problems on it. When she tries to put 7 problems on each page, the last page would only have 6 problems on it. Which of the following could be the total number of math problems?

24

29

35

209

499

Answer 1

Use the Chinese Remainder Theorem. We find that:
\[
x \equiv 4 \mod 5\\
x \equiv 5 \mod 6\\
x \equiv 6 \mod 7
\]So, it allows us to get (in about 23049872309487 different ways):
\[
x \equiv 209 \mod 210
\]