If b^2=a then b is what of a?

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
.