If x and y are positive integers and x is a prime factor of y^2, then which of the following MUST be true?A.
x is even
B.
x=y^2C.
\frac{y}{x} is even
D.
\frac{x}{y}\leq{1}C.
x+1 is also a prime factor of
y^2Given:
x=prime,
y=positive \ integer and
x is a prime factor of
y^2. Now, as
x is
a prime factor of
y^2 and
y is an integer then
x must be a factor of
y as well (how else could
x appear in the prime factorization of
y^2=y*y if it's not a factor of
y). Next, as
x is a factor of
y (and
y is a positive integer) then
x\leq{y} -->
\frac{x}{y}\leq{1}, so D must be true.
Another approach:Note that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions
if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.
A.
x is even --> not necessarily true:
x=3=odd and
y=3;
B.
x=y^2 --> never true, as
x=prime then it can not equal to square of another integer;
C.
\frac{y}{x} is even --> not necessarily true: if
x=3 and
y=3 then
\frac{y}{x}=1=odd;
D.
\frac{x}{y}\leq{1} is less than or equal to 1 --> always true as shown above or by POE;
E.
x+1 is also a prime factor of
y^2 --> not necessarily true: if
x=2 and
y=2 then
x+1=3 and 3 is not a factor of
y^2=4.
Answer: D.
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88% (02:37) correct
