Divide and Conquer
(This is one of a series of GMAT tips that we offer on our blog.)
The quantitative section of the GMAT features a heavy emphasis on the divisibility of numbers, as multiple questions will require division as a necessary arithmetic step, and others will require you to reduce fractions or note whether a particular number is a factor of another. Because of this, the ability to see divisibility in short order is extremely helpful for both speed and accuracy.
Many numbers feature “tricks” to assist test-takers as they determine divisibility (i.e. an integer is divisible by 5 if its units digit is 0 or 5), but in many cases those tricks can be more trouble to memorize than they are worth. (We cover some of the more-worthwhile tips in one of last year’s Tip of the Week entries.) For numbers that don’t feature a quick-recognition divisibility trick, a universal divisibility strategy is likely the fastest way to perform the operation:
When assessing whether a number is divisible by another, you can take advantage of the algebraic concept that a(b + c) = ab + ac. In numerical terms, you could say that 7(17) = 7 (10 + 7), or 70 + 49, which equals 119.
Knowing that, we can apply the same concept in reverse – while 119 may not look to be an easy-to-divide number (it’s not even, not divisible by 3, not divisible by 5…), we can test for divisibility by 7 without having to perform long division. Instead, we can try to break 119 apart by subtracting multiples of 7 that are easy to calculate:
119 (our starting value)
-70 (a known multiple of 7)
49 (the difference)
-49 (another known multiple of 7)
0 (there is no remainder, so 119 is, indeed, divisible by 7)
In short, to determine if a number is divisible by a potential factor, you can simply subtract multiples of that factor until you reduce the number to 0. If you can’t arrive at 0, the lowest positive value you can reach is the remainder, and the initial number is not divisible by the other.
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