Question

show that the square of any positive integer cannot be of the form 6 m + 2 or 6 m + 5 for any Integer​

Answer 1

Answer:

Let the positive integer be a and b=6.

According to Euclid's division lemma,

a=bq+r

Here, a=6q+4 where 0<and =r<b

So possible values of r are 0, 1,2,3,4,5

CASE1. when r=0,

a=4

a^2 =16

a^2 =6(2)+4

a^2 =6m+4..............equation1

CASE 2.r=1,

a=10

a=6m+4(m=1).................eq2

CASE3.a=6m+4(m=2).......

...........................

Similarly do for case4, 5 and 6.YOU WILL NOT GET a=6m+2 or 6m+5.

At last write down..

Hence, the square of any posit integer the cannot be of the form 6m+2 or 6m+5 for any integer m