Find the greatest positive integer that cannot be expressed as a multiple of , as a multiple of , or as a sum of a multiple of and a multiple of .

Source: mathcontest.olemiss.edu 3/24/2014

**SOLUTION**

What kind of numbers satisfy the requirements? We look at numbers that are not multiple of or not multiple of and try to express them as a sum of a multiple of and a multiple of . If we cannot, the numbers satisfy the requirements.

From on all numbers not ending in or can be expressed as a sum of a multiple of and a multiple of by the following rules.

If the number ends in , subtract and get a sum =

If the number ends in , subtract and get a sum =

If the number ends in , subtract and get a sum =

If the number ends in , subtract and get a sum =

If the number ends in , subtract and get a sum =

If the number ends in , subtract and get a sum =

If the number ends in , subtract and get a sum =

If the number ends in , subtract and get a sum =

is the greatest positive integer that cannot be expressed as a multiple of , as a multiple of , or as a sum of a multiple of and a multiple of .

**Answer**: