tirupatibalaji wrote:

X and Y are positive integers. If X is a prime factor of Y^2 , which of the following MUST be true?

(a) X is even

(b) X=Y^2

(c)Y/X is even

(d) X/Y is less than or equal to 1

(c)X+1 is also a prime factor of Y^2

Please need a good explanation for this.

Bunuel ...please help

Given: \(x=prime\), \(y=positive \ integer\) 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. y/x is even --> not necessarily true: if \(x=3\) and \(y=3\) then \(\frac{y}{x}=1=odd\);

D. x/y 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.

Hope it's clear.

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