Question

prove that the square of any Integer is either of the form 5m, 5 M + 1 or 5m + 4, for some integer m

Answer 1

Answer:

Step-by-step explanation:

a=bq+r

bq=5

r=1,2,3,4,

If r=0

a=bq+r

(5q+0)2

5q2+2 (5q)(0)+02

25q2+0+02

5 (5q2)

5m where m= 5q2

If r=1

(5q+1)2

5q2+2 (5q)(1)+(1)2

25q2+10q +1

5 (5q2+2q)+120

5m+1 where m= 5q2+2qq

If r=2q

(5q+2)2qq

25q2+2 (5q)(2)+(2)2

25q2+20q+4

5 (5q2+4q)+4

5m+4 where m= 5q2+4q

Answer 2

Answer:

Step-by-step explanation: