GMAT, GRE Number Properties I, II, III Must Be True Question

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 an 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




These kinds of questions usually ask which of the following must be true, or which of the following is always so that it is true no matter what set of numbers you choose to plug in. 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. That's the main strategy for solving must be true questions, in a nutshell. 

I. x+y is divisible by 4 

if x=3 and y=7then x+y=10

this is not divisible by 4. So this statement is not always true

II. xy has an even number of factors

only perfect squares have an odd number of factors, 

as xx and yy are distinct prime numbers then xy cannot be a perfect square and thus cannot 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+yx+y can be a perfect square, for example, if x=3 and y=13 then x+y=16 is a perfect square

so x+y can have an odd number of factors. So this statement is not always true;

Answer: B (II only)

Tips about the perfect square with an example:

1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and an EVEN number of Even-factors then it's a perfect square. For example, odd factors of 36 are 1, 3, and 9 (3 odd factors) and even factors are 2, 4, 6, 12, 18, and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example, 36=22∗32$36=2^2\ast 3^2$, powers of prime factors 2 and 3 are even.

NEXT:
There is a formula for Finding the Number of Factors of an Integer:

First make prime factorization of an integer n=ap∗bq∗cr$n=a^p\ast b^q\ast c^r$, where a$a$, b$b$, and c$c$ are prime factors of n$n$ and p$p$, q$q$, and r$r$ are their powers.

The number of factors of n$n$ will be expressed by the formula (p+1)(q+1)(r+1)$(p+1)(q+1)(r+1)$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: 450=21∗32∗52$450=2^1\ast 3^2\ast 5^2$

The total number of factors of 450 including 1 and 450 itself is (1+1)∗(2+1)∗(2+1)=2∗3∗3=18$(1+1)\ast (2+1)\ast (2+1)=2\ast 3\ast 3=18$ factors.

Back to the original question:

Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even --> let's say n=ap∗bq∗cr$n=a^p\ast b^q\ast c^r$, given that the number of factors of n$n$ is even --> (p+1)(q+1)(r+1)=even$(p+1)(q+1)(r+1)=even$. But as we concluded if n$n$ is a perfect square then powers of its primes p$p$, q$q$, and r$r$ must be even, and in this case number of factors would be (p+1)(q+1)(r+1)=(even+1)(even+1)(even+1)=odd∗odd∗odd=odd≠even$(p+1)(q+1)(r+1)=(even+1)(even+1)(even+1)=odd\ast odd\ast odd=odd\ne even$. Hence n$n$ cannot be a perfect square. Sufficient.

(2) The sum of all distinct factors of N is even --> if n$n$ is a perfect square then (according to 3) sum of odd factors would be odd and the sum of even factors would be even, so the sum of all factors of the perfect square would be odd+even=odd≠even$odd+even=odd\ne even$. Hence n$n$ cannot be a perfect square. Sufficient.

Answer: D