Are you ready for a challenge? Try to solve the following question in under two minutes:
How many different positive divisors does the number 147,000 have?
If you feel like two minutes are not nearly enough to solve the problem, you’re not alone. Even the most seasoned GMAT veterans might find the problem challenging, as it requires a deep level of understanding of two mathematical concepts: Divisibility and Combinatorics (just a fancy word for ‘counting’).
If I replaced the number 147,000 with the number 24, many more people would be able to come up with an answer:
You could just pair up the divisors (factors) and count them. Start with the extremes (1×24) and work your way in:
A quick count will show the number 24 has exactly 8 different positive divisors.
The number 147,000 will have many more positive divisors – too many to count… This is a strong indication that we will need to use combinatorics.
Divisibility: Any positive integer in the universe can be expressed as the product of prime numbers.
A few examples:
The number 24 is comprised of three factors of 2 and one factor of 3.
We should be able to ‘build’ any divisor of 24 using some combination of those prime factors; for example, we could choose to use none of them. What do you get if you use none of those prime factors?
Is 1 a divisor of 24? Absolutely!
What if we choose to use all of the prime factors that we have at our disposal?
Is 24 a divisor of 24? Of course!
I can use zero, one, two, or three factors of 2, and for each of those options, I can choose to use either zero or one factor of 3.
Combinatorics: When faced with several choices that are independent of each other, the total number of combinations will be the product of the number of choices in each decision point. For example, if you can choose among 3 appetizers, 2 mains, and 4 desserts when ‘building’ your 3-course meal, the number of different meals you could construct is 3 * 2 * 4 = 24
Intuitively, for each appetizer you could choose (out of 3), there are still 2 options for your main, and for each of those there are still 4 options for your dessert. As a rule of thumb, if you’re using the word each you should multiply the options.
Back to the number 24… allow me to repeat my last point here for your convenience: I can use zero, one, two, or three factors of 2 (four options total), and for each of those options, I can choose to use either zero or one factor of 3 (two options total).
Using combinatorics, we can now say there are exactly 4 * 2 =8 different positive divisors for the number 24.
Try to solve the original question again at this point, before you continue to read my solution.
There are four options for the number of 2 we will use (zero, one, two, or three), two options for the number of 3, four options for the number of 5, and three options for the number of 7.
In total, the answer to the original question is:
Is it just me, or is there a pattern here? You can just add 1 to each of the exponents of the prime numbers and then multiply them all together to get the total number of different positive divisors of any number!
This is a cool rule/formula to memorize, but as you guys know by now – memorizing little tricks and formulae will only get you so far on the GMAT. In order to crush it you really have to understand why these formulae work, and that was my goal with this blog post… Let me know in the comments whether you feel like you have a deeper level of understanding now!
- No matter how complex the problem you’re facing is, you can always simplify it by picking a nicer number (e.g. 24 rather than 147,000).
- Any positive integer in the universe can be expressed as the product of prime numbers.
- When faced with several choices that are independent of each other, the total number of combinations will be the product of the number of choices in each decision point.