Bethany wears jeans every 2 days. She wears running shoes every 3 days. If she wears jeans with running shoes on May 1, what are the next 3 dates on which she will wear both jeans and running shoes?

Bethany wears jeans on May 3rd, May 5th, and May 7th, and she wears running shoes on May 4th and May 7th. As a result, the next day Bethany wears both is 6 days later, on May 7th. We can continue this pattern to find that the next two such days are May 13th and May 19th, so the next three days Bethany wears both jeans and running shoes are May 7th, May 13th, and May 19th.

Notice that we have found the pattern that Bethany wears both jeans and running shoes every 6 days. 6=2*3, and she wears jeans every two days and running shoes every three days. Try figuring out why this is. Would this trick work with similar problems?