Josh has toy soldiers and needs to arrange them.

Question

Josh has more than 250 toy soldiers. when he tries to arrange them in rows of 3, there are 2 left over. When he tries to arrange them in rows of 5, there are 2 left over. When he tries to arrange them in rows of 7, there are 2 left over. What is the least number of toy soldiers Josh may have? 

Answer 1

Must be 2 more than a multiple of 3, 5, and 7 that's also greater than 250. The actual LCM is 105. 210 is still not enough, so it must be 315. With 2 left over, he has 317 toy soldiers.

Answer 2

The first step in solving this problem is find the least common multiple (LCM) of 3, 5 and 7 that is greater than 250. The easiest way to accomplish this is to prime factor each of these numbers. Since all three numbers are prime numbers, each number only has two factors, 1 and the number itself (1 & 3) (1 & 5) and (1 & 7). By multiplying 3 x 5 x 7 we get the product 105. The problem states the number of toy soldiers is greater than 250, therefore the LCM must be greater than 250. The next common multiple of the three numbers is 210, which is still too low. The next common multiple is 315. This number exceeds the 250 minimum number of soldiers requirement, so 315 is the LCM of 3, 5, and 7 that meets the problem's criteria. At this point since he always has two soldiers let over when he creates columns of 3, 5 or 7, we must add 2 to the 315. The answer is 317 toy soldiers.