tibukooinm
2022-04-01
kattylouxlvc
Beginner2022-04-02Added 11 answers
Now as you got and , we solve further by taking cases.
Therefore let for some positive integer k and by judging the equation we can see that p and q are even too. Let for some positive integers a and b. Substituting the values of m,p and q, we get
This implies
Now let's assume that n is odd, but that would mean is not divisible by 4, so either and r vice versa (Remember that a square is always .
Therefore in both cases. But we have on the other side which is always either or .
Thus, we get a contradiction. This implies n is even.
Now we assume n is not divisible by 3 i.e either or . Now a square is always either .
Therefore is never possible and thus the remaining possibility is . This implies and . Therefore, , but we had from the other equation which is always . Thus, we get a contradiction. Hence, n is divisible by 3 as well. Thus, n is divisible by 6.
Similar analysis goes for the other case where m is odd.
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