Suppose that n is a positive integer of the form 4m 3 where

Suppose that n is a positive integer of the form 4m + 3 where m element of Z. Explain why n must have a prime factor of the form 4k + 3 for k Elementof Z.

Solution

Any number of the form 4K+3 must be odd

so it cannot have any factors of the form 4k or 4k+2 which are even

Thus all the factors of a 4K+3 must be of the forms 4k+1 and 4k+3.

Suppose that they were all of the form 4k+1

multiplying two such yields (4k+1)(4m+1)= 4 (4km + k +m) + 1

Another 4k+1.

Thus the product of any number of factors of the form 4k+1 must be another 4k+1.

Thus a 4k+3 must have a prime factor of the form 4k+3.