Quarter Wit, Quarter Wisdom: The Power of Deduction on GMAT Data Sufficiency Questions

Quarter Wit, Quarter WisdomIn a previous post, we have discussed how to find the total number of factors of a number. What does the total number of factors a number has tell us about that number? One might guess, “Not a lot,” but it actually does tell us quite a bit! If the total number of factors is odd, you know the number must be a perfect square. If the total number of factors is even, you know the number is not a perfect square.

We know that the total number of factors of a number A (prime factorised as X^p * Y^q *…) is given by (p+1)*(q+1)… etc.

So, if we know that a number has, say, 6 total factors, what can we say about the number?

6 = (p+1)*(q+1) = 2*3, so p = 1 and q = 2 or vice versa.

A = X^1 * Y^2 where X and Y are distinct prime numbers.

Today, we will look at a data sufficiency question in which we can use factors to deduce much more information than what we might first guess:

When the digits of a two-digit, positive integer M are reversed, the result is the two-digit, positive integer N. If M > N, what is the value of M?

Statement 1: The integer (M – N) has 12 unique factors.
Statement 2: The integer (M – N) is a multiple of 9.

With this question, we are told that M is a two-digit integer and N is obtained by reversing it. So if M = 21, then N = 12; if M = 83, then N = 38 (keeping in mind that M must be greater than N). In the generic form:

M = 10a + b and N =10b + a (where a and b are single-digit numbers from 1 to 9. Neither can be 0 or greater than 9 since both M and N are two-digit numbers.)

We also know that no matter what M and N are, M > N. Therefore:

10a + b > 10b + a
9a > 9b
a > b

Let’s examine both of our given statements:

Statement 1: The integer (M – N) has 12 unique factors.

First, let’s figure out what M – N is:

M – N = (10a + b) – (10b + a) = 9a – 9b

Say M – N = A. This would mean A = 9(a-b) = 3^2 * (a-b)

The total number of factors of A where A = X^p * Y^q *… can be calculated using the formula (p+1)*(q+1)* …

We know that A has 3^2 as a factor, so X = 3 and p = 2. Therefore, the total number of factors would be (2+1)*(q+1)*… = 3*(q+1)*… = 12, so (q+1)*… must be 4.

Case 1:
This means q may be 3 so that (q+1) is 4. Since a-b must be less than or equal to 9 and must also be the cube of a number, (a-b) must be 8. (Note that a-b cannot be 1 because then the total number of factors of A would only be 3.)

So, a must be 9 and b must be 1 in this case (since a > b). The integers will be 91 and 19, and since M > N, M = 91.

Case 2:
Another possibility is that (a-b) is a product of two prime factors (other than 3), both with the power of 1. In that case, the total number of factors = (2+1)*(1+1)*(1+1) = 12

Note, however, that the two prime factors (other than 3) with the smallest product is 2*5 = 10, but the difference of two single-digit positive integers cannot be 10. This means that only Case 1 can be true, therefore, Statement 1 alone is sufficient. This is certainly not what we expected to find from just the total number of factors!

Statement 2: The integer (M – N) is a multiple of 9.

M – N = (10a + b) – (10b + a) = 9a – 9b, so M – N = 9 (a-b) . This is already a multiple of 9.

We get no new information with this statement; (a-b) can be any integer, such as 2 (a = 5, b = 3 or a = 7, b = 5), etc. This statement alone is insufficient, therefore our answer is A.

Don’t take the given data of a GMAT question at face value, especially if you are expecting questions from the 700+ range. Ensure that you have deduced everything that you can from it before coming to a conclusion.

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Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!