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An odd number added to itself an odd number of times yields

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by Vincen » Fri Sep 29, 2017 5:52 pm
An odd number added to itself an odd number of times yields

(A) an odd number
(B) an even number
(C) a prime number
(D) a positive number
(E) a perfect square

The OA is A.

I did it for a particular number, but is there any way to prove it in general?
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by Jay@ManhattanReview » Sun Oct 01, 2017 9:29 pm
Vincen wrote:An odd number added to itself an odd number of times yields

(A) an odd number
(B) an even number
(C) a prime number
(D) a positive number
(E) a perfect square

The OA is A.

I did it for a particular number, but is there any way to prove it in general?
Say an odd number is 3, and it is added 5 times (odd number of times) to itself (3).

Thus, the result = 3 + 3 + 3 + 3 + 3 = 15, and odd number.

An algebraic approach:

Say an odd number is n and it is added m times, where m is also an odd number.

Thus, the result = n + n + n + ... m times = m*n = Odd * Odd = Odd.

The question is not drafted well. The correct should be: An odd number added to itself an odd number of times must yield

Otherwise, except option B, all the options can also be correct, though not always. For example, if 1 is added 3 times, the result is a prime and a positive number. Similarly, if 3 is added 3 times, the result is a perfect square number.

Thus, except B, all option could be correct; however, only A must be correct.

Hope this helps!

-Jay

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by EconomistGMATTutor » Sun Oct 08, 2017 8:29 am
Vincen wrote:An odd number added to itself an odd number of times yields

(A) an odd number
(B) an even number
(C) a prime number
(D) a positive number
(E) a perfect square

The OA is A.

I did it for a particular number, but is there any way to prove it in general?
Hi Vincen,
Let's take a look at your question.

The question states that,
An odd number added to itself an odd number of times.
We know that an odd number can be represented as 2n + 1 in general.

An odd number added to itself 3 times:
(2n+1)+(2n+1)+(2n+1) = 3(2n + 1)

An odd number added to itself 5 times:
(2n+1)+(2n+1)+(2n+1)+(2n+1)+(2n+1) = 5(2n + 1)
...
An odd number added to itself an odd number of times let's say for (2n+3) times
(2n+1)+(2n+1)+(2n+1)+(2n+1)+...+(2n+1) = (2n+3)(2n+1)

We know that (2n+3) is an odd number and (2n+1) is an odd number.
The product of two odd numbers is an odd number always.
Therefore, (2n+3)(2n+1) is an odd number.

Option A is correct.

I am available if you'd like any followup.
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Re: An odd number added to itself an odd number of times yields

by Scott@TargetTestPrep » Sun Feb 02, 2020 10:15 am
Vincen wrote: ↑
Fri Sep 29, 2017 5:52 pm
 An odd number added to itself an odd number of times yields

(A) an odd number
(B) an even number
(C) a prime number
(D) a positive number
(E) a perfect square

The OA is A.

I did it for a particular number, but is there any way to prove it in general?
1 added to itself 3 times is:

1 + 1 + 1 = 3

3 added to itself 5 times is:

3 + 3 + 3 + 3 + 3 = 15

Thus, we see the result must be an odd number.

Answer: A

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