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