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 and are integers such that then either and or else and
Let us consider the following three cases.
1) If then and hence
2) If then implies Since we conclude that there is no such integer with
3) Finally, let , then either or If then If then
We conclude that only in the case 3 the equality is possible, and consequently, if and are integers such that then either and or else and