Question

Write five pair of integers (m, n ) such that m ÷ n = -3. One such pair is (-6, 2).

Answer 1

Answer:

Given a squarefree a and c>b>1 such that b is not a factor of c, then we can define m=ab2 need n=abc.

Then n=abc>ab2=m, and n2=a2b2c2 is divisible by m=ab2, and nm=abcab2=cb is not an integer since c is not divisible by b.

On the other hand, given any two m,n satisfying these conditions, we can find a,b,c.

If m is square-free, then m∣n2 implies m∣n. (Why?) So m is not square-free.

This means that we can write (uniquely) m=ab2 where a is square-free and b>1.

Now, since ab2∣n2, with a square-free, then then ab∣n. (Why?)

Writing n=abc for some c, then we then see n>m means c>b. We also see that that m=ab2 is a factor of n=abc if and only if b is a factor of c.

So this construction gives all m,n uniquely: Start with a square-free, and c>b>1 with c not a multiple of b. Then take m=ab2,n=abc.

Simple cases: When n=1 and b>1 we can choose c=b+1 then we get m=b2 and n=b(b+1).