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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