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Archangel Macsika

29. Prove or disprove that if m and n are integers such that mn = 1 then either m = 1 and n = 1 , or else m = - 1 and n = - 1 .

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Let us prove that if m and n are integers such that mn = 1 then either m = 1 and n = 1 , or else m = - 1 and n = - 1 .

Let us consider the following three cases.

1) If m=0 then mn=0\cdot n=0, and hence mn\ne 1.

2) If |m|>1, then mn = 1 implies |n|=\frac{1}{|m|}<1. Since n\ne 0, we conclude that there is no such integer n with 0<|n|<1.

3) Finally, let |m|=1, then either m=1 or m=-1. If m=1, then n=\frac{1}{m}=1\in\Z. If m=-1, then n=\frac{1}{m}=-1\in\Z.

We conclude that only in the case 3 the equality mn = 1 is possible, and consequently, if m and n are integers such that mn = 1 then either m = 1 and n = 1 , or else m = - 1 and n = - 1 .

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