Question

Prove that square of any positive integer are 5q,5q+1,5q+4

Answer 1

let 'a' be square of any positive integer

by Euclids division lemma

q=bq+r

0_<r<b

r<5

therefore possible outcomes are

• 5q(when r=0)
• 5q+1(when r=1)
• 5q+2(when r=2)
• 5q+3(when r=3)
• 5q+4(when r=4)

a²=(5q)²

a²=25q²

=5(5q)

where q is 5q

similarly you solve the rest and you will get the answer

plzz mark as brainiest

Answer 2

Here. on using Euclid division lemma

a=bq+r

• On taking arbitrary number b= 5
• So possibility of r=0,1,2,3,4
• If a = 5q
• On squaring it a=25q²
• a=5(5q²)
• = 5q where q is an integer.
• a=5q+1
• On squaring it a=25q² +10q +1
• a=5(5q²+2q)+1
• 5q+1 where q is an integer.
• a=5q+2
• On squaring it a=25q² +20q+4
• a=5(5q²+4q) +4
• 5q+4 where q is some integer
• a= 5q+3
• On squaring it a= 25q²+30q+9
• = 25q²+30q+5+4
• =5(5q²+6q+1)+4
• = 5q+4 where q is some integer.
• a=5q+4
• On squaring it a= 25q²+ 40q+16
• = 5(5q²+8q+3)+1
• = 5 q +1 where q is some integer.

Hence,the square of any integer is in the form of 5q, 5q+1,5q+4.