Question

1) Prove for any integer n, if n² is a multiple of 6 then so is n. To get credit, you should use the following facts in your proof:

If n² is even then so is n. (Proved)

If n² is a multiple of 3, then so is n. (Proved)

2)By contradiction, prove that the square root of 6 is irrational.

The result of part 1 should be be used as Lemma in your proof.

Answer 1

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since n² is multiple of 6 n² can be written k*6 where k is some integer since 2 divides 6, and 6 divides n² =. 2 divides n² so n²= x*2 where x is some integer => n² is even then so is n

since 3 divides 6, and 6 divides n² =. 3 divides n² = y*3 where y is some integer => n² is a multiple of 3, then so is n.

so n is multiple of 6