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There is no such number.

For suppose x was a number such that it keft the same remainder r, when divided by 99 and 204.

Then x = 99a + r

and x = 204b +r for some integers a and b.

Consider y = x + 99*204 then y > x

Now y = (99a + r) + 99*204 = 99*(a+204) + r

so y leaves a remainder of r when divided by 99.

Also y = (204b + r) + 99*204 = 204*(b+99) + r

so y leaves a remainder of r when divided by 204.

This contradicts the proposition that x is the maximum such number.

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Q: What is the largest positive integer that leaves the same remainder when divided into each 99 and 204?
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