Factor Faster

Last week, I wrote about how to avoid doing long division on GMAT math problems. One of the ways I suggested doing that was by factoring. It's one thing to say that, yet another to show how to do it; today's focus will be on learning to factor faster. The benefits? Skip the long division, divide faster, simplify fractions efficiently, and get a better grip on how numbers are related. There are few GMAT-related skills that will help you on a broader range of questions.

A Few Simple Rules

No matter how big a number is, it's simple to check whether it's a multiple of 2, 3, 4, 5, 6, 9, or 10. If you're going to factor quickly, you need to internalize these rules.

Divisible by 2? If the last digit of a number is even, the number is divisible by 2.

Divisible by 3? If the sum of the digits of a number is divisible by 3, the number itself is divisible by 3. For instance, the number 4,383 is divisible by three: the digits (4+3+8+3) sum to 18, which itself is divisible by 3.

Divisible by 4? If the last two digits of a number are divisible by 4, the number itself is divisible by 4. For example, the number 2,180 is divisible by 4 because 80 is divisible by 4.

Divisible by 5? Check the last digit of the number: if it's 5 or 0, the number is divisible by 5.

Divisible by 6? If a number is divisible by both 2 and 3, it's divisible by 6.

Divisible by 9? This is the same rule as 3, only the sum of the digits must be a multiple of 9. For instance, 4,383 (as shown above) is a multiple of 9, as well: the digits sum to 18, which is a multiple of 9.

Divisible by 10? If the last digit of the number is 0, the number is divisible by 10.

Other Rules

There are rules for 7, 8, and a slew of other small numbers. However, none are simple enough to be easily applicable under the pressure of the GMAT quant section. If you're interested in number theory, this is an interesting approach for determining divisibility by 7.

Now What?

Of course, the usual goal when finding factors of a number is to divide that number by one or more of those factors. Here are a couple of tips:

1. Start with small factors. It's easier to divide by 2 than by 6. With just a little practice, you can divide any number by 2 or 3 in your head. If a number is divisible by 4, it may be faster to divide by 2, then divide by 2 again than to divide by 4 once.
2. Look for nearby round numbers. If you needed to divide 441 by 3, keep in mind that 441 = 450 – 9. 450 divided by 3 is 150; 9 divided by 3 is 3. 150 – 3 = 147. Alternatively, 441 = 420 + 21. By the same reasoning, the results are 140 + 7 = 147. You can apply this method with any factor, though 3 are 4 tend to be the easiest.
3. Practice. The more mental math you do, the better you'll get at it. It's that simple. When paying bills, try balancing your checkbook in your head. (You might want to double-check when you're done!) At the store, figure out how much 3, 6, 9, or 12 of something would cost. The more time you spend thinking about numbers, the more comfortable you'll be and the faster you'll get.

Mental math will be one of the recurring themes of this site. There are few things you can do that will improve your efficiency and reduce your careless mistakes more than increasing your facility with simple calculations. They may seem to slow you down at first, but stick with it: eventually, the results will astound you.

About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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