Do you understand the connection between this problem and the first paragraph? Every integer can be written in one of forms:

5n, 5n+ 1, 5n+ 2, 5n+ 3, or 5n+ 4- in other words, every integer has remainder 0 to 4 when divided by 5.

Suppose x is a mutiple of 5: x= 5n. Then x^2+ 5y= 25n^2+ 5y= 5(5n^2+ y), a multiple of 5. But 243723 is NOT multiple of 5.

(5n+ 1)^2= 25n^2+ 10n+ 1= 5(5n^2+ 2n)+ 1 has remainder 1 when divided by 5.

(5n+ 2)^2= 25n^2+ 20n+ 4= 5(5n^2+ 4n)+ 4 has remainder 4 when divided by 5.

What about the others? What are the possible remainders, when divided by 5, of

?