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# Number Theory/HCF

Question
The HCF of (a-1)(a^3 + m) and (a+1)(a^3 - n) and (a+1)(a^2 - n) is a^2 - 1, find the values of m and n.

I just managed to get to my PC again, and here is my answer.

Note that the factors of a^2 - 1 are (a+1) and (a-1).
If each of the terms is divided by this, we get
(a^3 + m)/(a+1),
(a^3-n)/(a-1), and
(a^2-n)/(a-1).

For the 1st fraction be divisible by (a+1), m needs to be +1, for then the top factors in
(a+1)(a^2 - a + 1), so the result is a^2 - a - 1.

For the 2nd fraction to be divisible by a-1, the top needs to have n as 1.
The result of this would be (a^3-1)/(a-1) = a^2 + a + 1.

The same is true for the 3rd fraction, in that here again n needs to be 1.
The result is then (a^2 - 1)/(a-1) = (a+1)(a-1)/(a-1) = a+1.

Number Theory

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