What should the integer values of $l,m$ be so that $l(l+m)=8$.

Question

How to find the integral solutions of the equation $l(l+m)=8$,I think it should be $l=8,m=-7$ and $l=1,m=7$.But I am looking for a general method to solve such problem.Also I found that $l=-1,m=9$ and $l=-8,m=7$ are solutions.How to find all possible solutions.Is there anything related to prime numbers or number theory?

Answer 1

It is related to number theory.

The prime factorization of $8$ is $2^3$,

so the factors of $8$ are $\pm2^0=\pm1, \pm2^1=\pm2, \pm2^2=\pm4,$ and $\pm2^3=\pm8$.

Thus, $1×8, 2×4, 4×2, 8×1,−1×−8, −2×−4,−4×−2,$ and $−8×−1$

are possible ways $8$ can be written as a product of two integers.

Setting $l$ to be one of the factors and $l+m$ to be the other yields eight solutions to your problem.

Besides the four you found, there are $(l,m)= (2,2), (4,-2), (-2,-2), $ and $(-4,2)$.