Using differences of Squares

CAT Exam
Whenever we’re given unpleasant numbers on the CAT, it’s worthwhile to think about the characteristics of round numbers in the vicinity. Like, 3599 is the same as 3600 – 1. 3600, the beautiful round number that it is, is a perfect square: 602. And 1 is also a perfect square: 12. Therefore 3600 – 1 can be written as the following difference of squares: 3600 – 1 = 602 – 12 We know that x– y= (x + y)(x – y), so if we were to designate “x” as “60” and “y” as “1”, we’ll arrive at the following: 60– 1= (60 + 1)(60 – 1) = 61 * 59 Now we know that 61 and 59 are both factors of 3599. Because 3599 has factors other than 1 and itself, we’ve proven that it is not prime, and earned ourselves a plumb job at a hedge fund. Not a bad day’s work. But let’s not get ahead of ourselves. Let’s analyze some questions that incorporate this concept. First: 999,9992 – 1 =  A) 1010 – 2 B) (10– 2) 2    C) 10(106 -2) D) 10(105 -2) E) 10(106 -2) Notice the pattern. Anytime we have something raised to a power of 2 (or an even power) and we subtract 1, we have the difference of squares, because 1 is itself a perfect square. So we can rewrite the initial expression as 999,9992 – 12. Using our equation for difference of squares, we get: 999,9992 – 1 = (999,999 +1)(999,999 – 1) (999,999 + 1)(999,999 – 1) = 1,000,000* 999,998. Take a quick glance back at the answer choices: they’re all in terms of base 10, so there’s a little work left for us to do. We know that 1,000,000 = 10 (Remember that the exponent for base 10 is determined by the number of 0’s in the figure.) And we know that 999,998 = 1,000,000 – 2 = 106 – 2, so 1,000,000* 999,998 = 10(106 -2), and our answer is E. Let’s try one more: Which of the following is NOT a factor of 38 – 28? A) 97 B) 65 C) 35 D) 13 E) 5 Okay, you’ll see quickly that 38 – 28 will involve same painful arithmetic. But thankfully, we’ve got the difference of two numbers, each of which has been raised to an even exponent, meaning that we have our trusty difference of squares! So we can rewrite 38 – 28 as (34)2 – (24)2. We know that 34 = 81 and 24 = 16, so (34)2 – (24)2 = 812 – 162. Now we’re in business. 812 – 162 = (81 + 16)(81 – 16) = 97 * 65. Right off the bat, we can see that 97 and 65 are factors of our starting numbers, and because we’re looking for what is not a factor, A and B are immediately out. Now let’s take the prime factorization of 65. 65 = 13 * 5. So our full prime factorization is 97 * 13 * 5. Now we see that 13 and 5 are factors as well, thus eliminating D and E from contention. That leaves us with our answer C. Not so bad. Takeaways:
  • The CAT is not interested in your ability to do tedious arithmetic, so anytime you’re asked to find the difference of two large numbers, there is a decent chance that the number can be depicted as a difference of squares.
  • If you have the setup (Huge Number)2 – 1, you’re definitely looking at a difference of squares, because 1 is a perfect square.
  • If you’re given the difference of two numbers, both of which are raised to even exponents, this can also be depicted as a difference of squares, as all integers raised to even exponents are, by definition, perfect squares.

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