## Arithmetic Concepts in GMAT Quant

# How to spot prime factorisation when it isn't explicitly referenced

Although most students know how to determine the prime factorisation of a number, only rarely will you see a question phrased so bluntly. More likely, the question will hint at the issue in a roundabout fashion:

Ifx,yandzare positive integers, is it true that x^{2}is divisible by 16?

- 5x=4y
- 3x
^{2}=8zA. Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

B. Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are not sufficient.

There’s not a word mentioning prime factorisation, yet this is the skill being tested. Two hints clue us in to this:

- The test for
**divisibility**involves**variables**, but**no remainder**is mentioned - All (or many) of the variables in the problem are
**limited to integers**

The presence of these two clues together often suggest that you should divide and conquer.

Let's start with statement 1):

5x=4y

Divide both sides by 4 to get:

(5x) / 4 = y

If we now focus only on the left-side of the equation, we only need to concern ourselves with a single expression with a single variable: 5*x*/4.

In order for 5*x*/4 to be an integer, all of the prime factors in the denominator (two 2′s) must also be present in the numerator (which contains 5 and x). This must be true in order for 5*x*/4 to be an integer. Because the 5 in the numerator does not contain any factors of 2, *x* must contain at least two factors of 2.

Therefore, x^{2} must contain at least four factors of 2. In other words, x^{2} must also divisible by 16. The answer to the question stem is therefore a definite **yes**.

Statement 2) can be conquered using the same tactic:

3x^{2}=8z

Divide both sides by 8 to obtain:

(3x^{2}) / 8 = z

Using the same logic, we see that 3x^{2} must contain at least three factors of 2. Since 3 obviously won’t contain any factors of 2, they must all come from the x^{2} term. At first glance, it may seem like we have insufficient data, since we have only been able to prove that x^{2} has at least three factors of 2 rather than four. Notice, however, that since x itself is an integer, x^{2} must be a perfect square.

**A perfect square must contain an even number of each prime factor.** As an example, consider the perfect squares 4 (two factors of 2), 81 (four factors of 3), and 400 (four factors of 2, two factors of 5). If x^{2} contained only three factors of 2, such as if x^{2}= 8, it would no longer be a perfect square. If we were to take √(x^{2}), we would end up with *x* = √8 = 2.82, which is not an integer. As a result, x^{2} must always contain an even number of factors, or a minimum of at least four factors of 2. Once again, we have a definite** "Yes"** answer to the question stem. The answer to this data sufficiency question is therefore **D): each statement alone is sufficient.**