Breaking Down a GMATPrep Divisibility Problem
This week, were going to tackle a challenging GMATPrep problem solving question from the topic of Divisibility (a subset of Number Properties).
Lets start with the problem. Set your timer for 2 minutes. and GO!
*If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?(1) x = 12u, where u is an integer
(2) y = 12z, where z is an integer
The first thing youll probably notice: I didnt include the answer choices. The five Data Sufficiency answer choices are always the same, so we should have those memorized. If you dont have them memorized yet, add this to your to do list.
Just in case, here are the five choices (in casual language, not official language):
(A) statement 1 works but statement 2 does not work
(B) statement 2 works but statement 1 does not work
(C) the statements do NOT work alone, but they DO work together
(D) each statement works by itself
(E) nothing works, not even using them together
Okay, now that weve got that out of the way, lets tackle this problem! What is a greatest common divisor?
A plain divisor is a number that divides evenly into another number (divides evenly = returns an integer answer). This kind of number is also called a factor; divisor is just a synonym for factor. For example, 3 is a divisor of 12, because 12 divided by 3 equals an integer (4). Whats another divisor of 12? What is 12 a divisor of? (1, 2, 3, 4, 6, and 12 are all divisors of 12. 12 is a divisor of 24, 36, 48, and so on.)
A common divisor can be calculated for two or more numbers. For example, 3 is a divisor of 12 and its also a divisor of 15. 3, then, is a common divisor of 12 and 15.
The greatest common divisor is the largest number in the set of common divisors between two (or more) numbers. 3 is a common divisor of 12 and 15. Is it the greatest common divisor?
You could test that by writing down all of the divisors for 12 and 15 and then comparing. 12s divisors are 1, 2, 3, 4, 6, and 12. 15s divisors are 1, 3, 5, and 15. 3 is the largest number in common, so yes, 3 is the greatest common divisor of 12 and 15.
Theres one more important thing to discuss here before we dive into the problem important but complex. Ill illustrate with numbers first and then well derive a general principle. 12 and 15 are three digits apart. 3 is also a common divisor of 12 and 15. Three digits apart and 3 is the common divisor by definition, then, 3 must be the greatest common divisor of the two! Consider the numbers 15 and 20. These two are five digits apart and 5 is a common divisor of the two. 5, then, is the greatest common divisor. Think about why before you keep reading.
So, weve said that if the two numbers are a certain number of digits apart, and those two numbers are also divisible by that number of digits, then that number of digits equals the greatest common divisor. In order to prove this to yourself, think about what would be necessary in order for this NOT to be true. Weve got 12 and 15, which are separated by 3 digits and which share 3 as a common divisor, but we want to argue that 3 is not the greatest common divisor (GCD). Whats to stop 5 (a divisor of 15) or 6 (a divisor of 12) from being the GCD? (Besides the fact that 12 isnt divisible by 5 and 15 isnt divisible by 6!)
Well, if 5 is a divisor of two numbers, then those two numbers are both multiples of 5. Given that the two numbers are not the same number, what is the minimum distance you can have between the two? Ah, right. They have to be at least 5 digits apart. What about 6? Same thing. The two different multiples of 6 have to be at least 6 units apart. 12 and 15 are only three units apart so they cant have a common divisor larger than three.
Okay, now that you know that, go back and examine this problem again. Do you want to keep your initial answer or change it?
Were given that x and y are positive integers, and were asked to find the greatest common divisor of these two. Also, were told that x = 8y + 12.
Thats interesting. 8y has to have 2 as a factor (because of the 8), and 12 also has 2 as a factor, so the sum of those two numbers, x, must have 2 as a factor. In fact, 8 and 12 also share 4 as a factor, so x must also have 4 as a factor. The general rule is:
multiple of 2 + multiple of 2 multiple of 2, or adding multiples of 2 will produce another multiple of 2
Note that I didn't use an equals sign there, but an arrow; it only works in that direction. Any other number can be inserted into the above; it will still hold true. Also, that addition sign can be changed to a subtraction sign. In short, if we add or subtract two numbers with a common factor, the sum or difference will also share that common factor.
I dont know anything else about x and y, though, so I dont know what the greatest common factor is. Lets look at the statements. First, were told that x = 12u, where u is an integer. That must be useless, because it doesnt say anything about y. Right?
Not so fast. Weve got more work to do before we can tell! From this statement, we know that 12 must be a factor of x (both 12 and u are factors of x, because they multiply to produce x). Combine this with the equation given in our question stem:
x = 8y + 12
[multiple of 12] = 8y + [multiple of 12]
Hmm. I could rearrange to this:
[multiple of 12] - [multiple of 12]= 8y
So that tells me that 8y is a multiple of 12. What does that tell me about y? Not enough, unfortunately. The full term 8y is divisible by the prime factors of 12 (2, 2, and 3). Is y by itself also divisible by all three of those factors? We dont know for sure. The two 2 factors could come from the constant 8, leaving only the 3 factor definitely coming from the y variable. We know y is divisible by 3, but thats all we know for sure. Statement 1 is not sufficient.
Statement 2 says that y = 12z, where z is an integer. This tells me y is a multiple of 12. Plug that one back into the equation given in the question stem:
x = 8[multiple of 12] + [multiple of 12]
Great! So x is also a multiple of 12, according to that rule we discussed earlier (add two multiples of 12 and the sum will also be a multiple of 12)! So 12 is a common divisor of x and y, but is it the greatest common divisor?
Yes, it is. The equation is really this:
x = [multiple of 12] + 12
That means that, whatever that first multiple of 12 is, were adding only 12 more to it. The two numbers x and 8y, then, are exactly 12 digits apart. Remember our other rule from above: if the two numbers are a certain number of digits apart, and theyre both divisible by that number, then that number is the greatest common divisor. In this case, x and 8y are both divisible by 12, and the two numbers are also 12 digits apart, so 12 is the greatest common divisor.
But we were asked about x and y, not 8y! Hmm. 8y is a multiple of y, so any factors of y are also factors of 8y (plus 8y has an additional three 2s). This means that the GCD of x and y has to be equal to or smaller than the GCD of x and 8y. (If youre not sure why, test some small numbers to see how it works.) We know that 12 is a factor of y, and 12 is also the GCD of x and 8y, so 12 must also be the GCD of x and y.
Statement 2 is sufficient; the correct answer is B.
Key Takeaways for Solving Hard Divisibility Problems:
- Determine that you have a divisibility problem. The problem might mention words such as divisible, divisor, factor, multiple, prime factor, greatest common factor (or divisor), or least common multiple.
- For harder problems, know the more obscure rules. When you add or subtract two numbers that share a common factor, the sum or difference also shares that factor. When you have two numbers that are n digits apart, and both numbers are also divisible by n, then n is the greatest common divisor, or greatest common factor, of the two numbers.
- Know how to use the above obscure rules in an abstract or theoretical way; you may not be given real numbers (as in the above problem).
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.