Question

show that square of positive integer can't be of the form 5m+2 or 5m+3​

Answer 1

Step-by-step explanation:

==> a = 5( 5q² + 8q + 3 ) + 1 . °• a = 5m + 1 . [ Where m = 5q² + 8q + 3 ] . → Therefore, square of any positive integer in cannot be of the form 5m + 2 or 5m + 3

Answer 2

Step-by-step explanation:

Let a be any positive integer

By Euclid's division lemma

a=bm+r

a=5m+r (when b=5)

So, r can be 0,1,2,3,4

Case 1:

∴a=5m (when r=0)

a²=25m²

a²=5(5m² )=5q

where q=5m²

Case 2:

when r=1

a=5m+1

a² =(5m+1)²=25m²+10m+1

a² =5(5m² +2m)+1

=5q+1 where q=5m²+2m

Case 3:

a=5m+2

a²=25m²+20m+4

a²=5(5m²+4m)+4

5q+4

where q=5m²+4m

Case 4:

a=5m+3

a²=25m²+30m+9=25m²+30m+5+4

=5(5m²+6m+1)+4

=5q+4

where q=5m²+6m+1

Case 5:

a=5m+4

a² =25m²+40m+16=25m²+40m+15+1

=5(5m²+8m+3)+1

=5q+1 where q=5m²+8m+3

From these cases, we see that square of any positive no can't be of the form 5q+2,5q+3.