When is the product of 2 nonzero integers less than or equal to both of the 2 factors?

Call the two factors x and y.  We do casework on whether x and y are positive or negative.

Case 1:  x and y are both positive.

We solve the equations:

x*y<=x

x*y<=y

Dividing the first by x and the second by y, we see that x<=1 and y<=1.  Since x and y must be positive integers, we have a solution at x & y = 1.

Case 2: One of x and y is negative, the other is positive.

Assume that x is the negative one (as we haven't assumed anything else about x, this doesn't affect our answer).

x is a negative value, and y is what we will be multiplying that negative value by.  If y is under 1, the negative value will decrease, which makes the overall value of the product bigger than x.  If it is at least 1, the negative value stays the same or increases, so the value of the product is less than or equal to x.  As y will always be at least 1 (because it's a positive integer), all values in this case work.  So, we have a solution for any positive integer and any negative integer.

Case 3:  Both x and y are negative.

In this case, x*y will be positive.  This is clearly greater than both x and y, so no values in this case work.

So, to summarize, our working values are (1, 1) and any positive integer combined with any negative integer.

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