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

Question

Mathematics

Winston Gallegos

27 April, 01:08

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

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Answers (1)

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Strong · 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.