IF THE SQUARE ROOT OF P2 IS AN INTEGER, P IS A PERFECT SQUARE. LET’...
4. If the square root of p
2
is an integer, p is a perfect square. Let’s
take a look at 36, an example of a perfect square to extrapolate
some general rules about the properties of perfect squares.
Statement I: 36’s factors can be listed by considering pairs of
factors (1, 36) (2, 18) (3,12) (4, 9) (6, 6). We can see that they are
9 in number. In fact, for any perfect square, the number of factors
will always be odd. This stems from the fact that factors can always
be listed in pairs, as we have done above. For perfect squares,
however, one of the pairs of factors will have an identical pair, such
as the (6,6) for 36. The existence of this “identical pair” will always
make the number of factors odd for any perfect square. Any number
that is not a perfect square will automatically have an even number
of factors. Statement I must be true.
Statement II: 36 can be expressed as 2 x 2 x 3 x 3, the product of 4
prime numbers.
A perfect square will always be able to be expressed as the product
of an even number of prime factors because a perfect square is
formed by taking some integer, in this case 6, and squaring it. 6 is
comprised of one two and one three. What happens when we square
this number? (2 x 3)
2
= 2
2
x 3
2
. Notice that each prime element of 6
will show up
twice in 6
2
. In this way, the prime factors of a perfect
square will always appear
in pairs, so there must be an even
number of them. Statement II must be true.
Statement III: p, the square root of the perfect square p
2
will have
an odd number of factors if p itself is a perfect square as well and
an even number of factors if p is not a perfect square. Statement III
is not necessarily true.
The correct answer is D, both statements I and II must be true.