1:: If a is divisible by a square of a prime number, then we cannot conclude from |2 a | b 2 that | a | b .
Any time a is divisible by a square of a prime number, =⋅ a = k ⋅ p n where ≥2 n ≥ 2 and p does not divide k , you can see that a divides (⋅−1)2 ( k ⋅ p n − 1 ) 2 , but a does not divide ⋅−1 k ⋅ p n − 1 .
Part 2: If a is not divisible by a square of a prime number, then we can conclude from |2 a | b 2 that | a | b .
On the other hand, if a is not divisible by a square of a prime number, then =12⋯ a = p 1 p 2 ⋯ p n where p i is prime for all p . Then, assuming a divides 2 b 2 , you can conclude that p i divides 2 b 2 , and therefore p i divides b (from prime factorization) for all i . From this, you can conclude that a divides b .
If we have an equation and we want to find what b is in relation to a, we can change the equation so that we have b on one side and whatever is on the other side is what b is.
To isolate b, we can take the square root of both sides as taking the square root of something squared results in the base.
