Question

Prove that the square of any positive integer is of the form 5q 5q+1 5q+4 or 5q + 9 for some integer q

Answer 1

smaller no. (b)=5
remember(r)=1,2,3
a^2=(bq+r)^2
=(5q+1)^2
=(5q)^2+2×5q+(1)^2=25q^2+10q+1
=5(5q^2+2q)+1
=5q+1(where q=5q^2+2)
prove
r=2
=(5q+2)^2
=25q^2+20q+4=5(5q^2+4q)+4=5q+4(where q=5q^2+4q)
proved
let's r=3
=(5q+3)^2
=25q^2+30q+9=5(5q^2+6q)+9
=5q+9(where q=5q^2+6q)proved

Answer 2

Answer:

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