## factorization and 4

Hi,

Can anyone confirm for me whether it has been proven that:

if a number is congruent to 1 mod 4 and is expressed as the product of two factors, the difference between those factors will always be congruent to 0 mod 4; and that if the number is congruent to 3 mod 4 the difference between two factors is congruent to 2 mod 4.

If it has been proven, or I suppose even researched, what do I google to read up on it?

I can't find a counterexample and I can't figure out why it is and it's driving me nuts. Why does 4 have this predictive property and other divisors don't?

Thanks much.
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 Recognitions: Gold Member In the case of ab==1 Mod 4, we need only consider the cases. If a==2 or b==2, then the product can not be congruent to one. So in the remaining cases, all that is possible is that a==b and both are congruent to 1 or 3. It's an easy problem.
 So you are saying it's because 4 is a small number with a limited number of cases that just happen to work out that way? I.e., no larger number has the same property because it's too large? Is this property for 4 related to it being even? Square? A power of two? Or is there no general theory on what limitations congruences place on factors? Sorry to be so thick.

Recognitions:
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