Why is the product of a rational number and an irrational number irrational?

The product of a nonzero rational number and an irrational number is irrational: if the product were rational, dividing it by the nonzero rational factor would make the irrational number rational, which is impossible. Irrational numbers have non-terminating, non-repeating decimal expansions, so choice A gives the intended reason. Note that the claim needs the rational number to be nonzero; multiplying by 0 gives 0, which is rational.

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