GMAT Tip of the Week: Breaking Down Factors With Breaking Bad

America has been buzzing for weeks about the last season of Breaking Bad, and the echo effect has taken hold even after this past Sunday’s finale as thousands rush to catch up on Netflix or DVD to get into the hype.

But regardless of where you are in the series, it’s important that you hear this one Breaking Bad spoiler:

Walter White would absolutely kill the GMAT.

For those of us still a season behind (but catching up rapidly) we don’t know yet whether Walter can outsmart the DEA, the Mexican cartel, or the New Mexican cartel (or even Jesse or Skyler for that matter) but we do know that he’d perform extremely well on the GMAT.

Why?

In large part because the GMAT has a strong conceptual emphasis on factors, multiples, and prime numbers, and because one of the best ways to understand the concept of prime factors is to see them like a chemist would.

Take the number 12 and the substance ‘water’ for example. Water is H20 = two hydrogen atoms and one oxygen atom. We identify water by H20 because hydrogen and oxygen are as far as you can really break water down without splitting atoms (which isn’t required on the GMAT). Once you’ve taken water all the way down to the atomic level, you know exactly what it takes to make water – two hydrogen atoms and one oxygen atom.

So, for example, if you had 75 hydrogen atoms and 20 oxygen atoms, how many molecules of water – H2O – could you make? You’re limited by the 20 oxygens, so 20 water molecules. Take those 20 oxygens, pair each one with two hydrogens (for a total of 40 hydrogens), and you’ll have enough for 20 water molecules with 35 “free” hydrogen molecules left over.

And in a more complicated example, say you had 5 molecules of propane (C3H8 – 3 carbons and 8 hydrogens per molecule) and 6 molecules of carbon dioxide (C02 – one carbon, two oxygens), how many molecules of water could you make from that mixture (obviously assuming you could break those bonds, etc.). You’ll want to first determine how many molecules of each you have: 5*8 of hydrogen, so 40 hydrogens, and 6*2 of oxygen, so 12 oxygen. So here we have plenty of hydrogen – enough for 20 molecules of water – but only 12 oxygen molecules, so we can only make 12 molecules of water with a bunch of carbon and hydrogen left over.

If you get that about chemistry, you can use that analogy to think about the number 12 differently. We can break a number like 12 down into its “atomic” components, too, via division. 12 is 6 * 2 or it’s 3 * 4, but either way if you keep dividing until you only have prime numbers, you get it down to 2 * 2 * 3, or 2^2 * 3. When you’re talking about factors, multiples, and divisibility, prime numbers play the role of atoms, and instead of H20 you use 2^2 * 3. But the concepts work quite similarly.

Take this question, for example – how many times can you divide the number 12! by 12 and still have an integer remaining?

Much like our water question above, this one can be solved by using an atomic/prime breakdown. To divide by 12, as we know, we need two 2s and a 3. So if we break out 12! into:

12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

Our goal, like we did with hydrogen and oxygen above, is to get our 2s and 3s out where we can combine them into 12s. That number line above then becomes:

(2 * 2 * 3) * 11 * (2 * 5) * (3 * 3) * (2 * 2 * 2) * 7 * (2 * 3) * 5 * (2 * 2) * 3 * 2 * 1

That’s ten 2s and five 3s, and we need two 2s and a 3 for each 12, so we can make five 12s.

The GMAT loves questions that deal with factors and multiples in this way – they’re conceptual, they don’t lend themselves well to brute-force calculations – and so being able to think in terms of prime factors is a very important skill. And for the chemistry-inclined, thinking of prime numbers as atoms is a helpful analogy. Factors and multiples work a lot like atoms and molecules – you can combine prime factors in many ways, but astute GMAT “chemists” see the ability to break apart those factors to see what they’re *really* looking at at the prime factor level. So if you’re chasing Harvard crimson or Stanford cardinal the way that Hank and the DEA were chasing blue crystal, borrow a tactic from Walter White and think about the chemistry of factors and multiples.

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By Brian Galvin

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