Question

Show the square of a positive integer is of the form 5m ,5m+1 or 5m+4​

Answer 1

a=(5m) or (5m+1) or (5m+2) or (5m+3) or (5m+4)

a²=25m² or 25m²+10m+1 or 25m²+20m+4 or 25m²+30m+9 or 25m²+40m+16

=5(5m²) or 5(5m²+2m)+1 or 5(5m²+4m)+4 or 5(5m²+6m+1)+4 or 5(5m²+8m+3)+1

=5m or 5m+1 or 5m+4 or 5m+4 or 5m+1

PROVED

Answer 2

Step-by-step explanation: According to Euclid's division lemma;given two positive integers a &b there exist a unique integer q &r where a=bq+r and 0≤r<b

ACC TO QUESTION:a=bq+r        b=5   r=0,1,2,3,4

CASE-1

when r=0

a=bq+r

a=5q+0=5q

a²=(5q)²

a²=25q²

a²=5(5q²)=

a²=5m where m=5q²

similarly

when r=1

a=5q+1

a²=(5q+1)²

a²=5q²+10q+1  (by identity (a+b)²}

a²=5(q²+2q)+1

a²=5m+1 where m=q²+2q

case-3

prove it like the one done above except reminder=3 and in case-4 reminder=4

hope this answer helped you