Q. If x and y are distinct prime numbers, each greater than 2, which of the following must be true?
(I) x+y is divisible by 4
(II)x * y has even number of factors
(III)x+y has an even number of factors
A. I only
B. II only
C. I and III only
D. II and III only
E. I and II only
I plugged in numbers for each option, but it took me more than 2 mins to get to the right answer -OA is B.
I had to consider x=13 and y=23 for ruling out (III), since x+y in this case =36 which has odd no. of factors.
PLease explain if there is any other method for solving this under 2 mins
First of all we are asked "which of the following MUST
be true" not COULD be true.
"MUST BE TRUE" questions:
These questions ask which of the following MUST
be true, or which of the following is ALWAYS
true no matter what set of numbers you choose. Generally 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.
As for "COULD BE TRUE" questions:
The questions asking which of the following COULD
be true are different: if you can prove that a statement is true
for one particular set of numbers, it will mean that this statement could be true and hence is a correct answer.
I. x+y is divisible by 4 --> if x=3
, which is not divisible by 4. So this statement is not always true;
II. xy has even number of factors --> only perfect squares have an odd number of factors (check this: a-perfect-square-79108.html?hilit=perfect%20square%20reverse#p791479
), as x
are distinct prime numbers
can not be a perfect square and thus can not have an odd number of factors, so xy
must have an even number of factors. This statement is always true;
III. x+y has an even number of factors --> now, x+y
can be a perfect square, for example if x=3
then x+y=16=perfect \ square
, so x+y
can have an odd number of factors. So this statement is not always true;
Answer: B (II only).
Hope it's clear.