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Square of odd number

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by GmatKiss » Sun Oct 16, 2011 12:53 pm
Which of the following is two more than the square of an odd integer?

(A) 14,173
(B) 14,361
(C) 14,643
(D) 14,737
(E) 14,981

Looking for a strategy to solve this one!
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by neelgandham » Sun Oct 16, 2011 1:43 pm
GmatKiss wrote:Which of the following is two more than the square of an odd integer?

(A) 14,173
(B) 14,361
(C) 14,643
(D) 14,737
(E) 14,981

Looking for a strategy to solve this one!
Answer - C)

How ?

On the first glance, I found that all the numbers are on either side of 14400(Square of 120) on the number line. Now, 119, 121 and 123 are the odd integers nearest to 120. I squared them - 14161, 14641, 15129.Adding 2, I got 14163, 14643, 15131.

Hence, Option C.

Time: 30 - 90 secs
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by Luke.Doolittle » Sun Oct 16, 2011 2:18 pm
I love the approach above for its insight!

Just for kicks here is another way to solve it. The primary tactic necessary is to know that an odd integer can be rewritten as 2k + 1 where k is an integer. If you square the odd integer you get:

(2k + 1)(2k + 1) = 4k^2 + 4k + 1 = 4(k^2 + k) + 1

Thus an odd integer squared is one more than a number divisible by 4. The problem asks which of the following is a number that is 2 more than the square of an odd integer. Thus the question could be rephrased as which of these numbers is 3 more than a number divisible by 4. If you know your divisibility properties you know that for a number to be divisible by 4 its last 2 digits must be divisible by 4 so we need a number whose last 2 digits, minus three, are divisible by 4. Going down the list

A) 70 (not divisible by 4)
B) 58 (not divisible by 4)
C) 40 (DIVISIBLE BY 4, STOP)

(C) is your answer!
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