# 2000 PMWC Problems/Problem T1

## Problem

A box contains to candies. When the candies are evenly distributed to , , , , or children, there is always one candy left. If the candies are packaged into small bags each having the same number of candies, what is the largest number of candies below in each bag so that no candies are left?

## Solution

If there is always one candy left over when the candies are evenly distributed to , , , , or children, then there must also be exactly one candy left over when the candies are distributed to children. Since , the total number of candies can be expressed as , where is any nonnegative integer. However, we know that there are between and candies, so the only viable value for is , meaning the total number of candies is .

The problem then asks us to find the largest number of candies below such that the candies are evenly distributed and no candies are left over. This is equivalent to asking what the largest factor of is that is less than . Since , the only factors of are , , and . Thus the answer is .

## See Also

Back to test: https://artofproblemsolving.com/wiki/index.php/2000_PMWC_Problems