Prove that for all integers a and b if a b then a2 b2Solu

Prove that for all integers a and b , if a | b then a² | b²

Solution

Given that a and b is integers, and

a|b = a divides b

means, a|b = c , where c is an integer,

Now

a^2|b^2 = a^2 divides b^2

a^2|b^2 = a*a/b*b = (a|b)*(a|b)

a^2|b^2 = c*c = c^2

we know that c is an integer, So c^2 will also be an integer,

Now,

a^2|b^2 = integer, which means that a^2 divides b^2

Let me know in the comments, if you have any confusion.