Question: Consider the statement, For all integers a and b , if the product of a and b is even, then a is even or b

Consider the statement, "For all integersaandb, if the product ofaandbis even, thenais even orbis even."What would be the beginning of a direct proof of this statement?

Assume thatais even orbis even. Thena or b= 2kfor some integerk.We must show that eitherais even orbis even, so it suffices to show one or the other.

Assume thatabis even. Thenab= 2k+1 for some integerk.We must show that eitherais even orbis even, so it suffices to show one or the other.

Assume thatabis even. Thenab= 2kfor some integerk.We must show that eitherais even orbis even, so it suffices to show one or the other.

Assume thatabis even. Thenab= 2kfor some integerk.We must show thatais even andbis even.

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