Question

Prove by contradiction that there do not exist integers m and n such that 14m+21n=100

Answer 1

You can factor the left hand side to get
14m+21n = 100
7*(2m+3n) = 100
7*x = 100
where x = 2m+3n is an integer

There are no solutions to 7x = 100 since 7 is not a factor of 100 (ie 7|100 is false). So that means there are no solutions to the original diophantine equation.