If a team scores only touchdowns and field goals, what scores cannot be achieved if the team scored under 45 points?

Question

In a recent football game a team scored either touchdowns (for 6 points each), or field goals (for 3 points each). If a team scores only touchdowns and field goals, what scores cannot be achieved if the team scored under 45 points?

Answer 1

If the team scores only field goals, possible scores are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42

 

If the team scores only touchdowns, possible scores are 6, 12, 18, 24, 30, 36, 42

 

This shows that any possible score reachable with a combination of touchdowns and field goals is achievable with only field goals (a touchdown is equivalent to two field goals).

 

The scores not achievable are 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44