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.
look at the number to the right of the decimal. if the number is higher than or is 5 round the number up, but if the number is less than 5 it will stay the same.