If 108 is a factor of N^{2} then which of the following could be the r

Question

If 108 is a factor of N^{2} then which of the following could be the remainder when N is divided by 36?

I. o
II. 18
III. 6

A. I only
B. III only
C. I& II only
D. I,II& III
E. None of these

KSBGC · Accepted Answer

108 is a factor of N^2

Prime factorization will give us the hint about the size of N. The we will divide it by 36.

108 = 2*54 = 2*6*9 = 2*3*2*3*3.

we have :

2^2

3^3

N^2  is a perfect square . thus . there must be at lest another 3. we need even power to make a integer perfect.

2^2*3^4 . we have to bring another 3 to make  3^4 .

Minimum value of N^2 = 2^2 * 3^4 = 324

18^2 = 324 .

So, N must be 18 in this case.

18/36 = 0 + 18 . 18 is the remainder.

but we know minimum value of  N^2  is 324.  N^2  could be even more bigger. These extended values will be the multiple of 324 and in this case 324 will be considered as N.

324/36 = 9 + 0

Remainder is 0.

Thus, C is the correct answer.

KanishkM · Answer

108 is a factor of  N^2  will mean that it will be fully consumed

108 = 2*2*3*3*3
 N^2  = 2*2*3*3*3*3 *k
N = 2*3*3*k

36 = 2*2*3*3

This means that Minimum value of N can be 2*3*3 = 18
18/36, remainder = 18

Now k can take some other value as well but it will be in a pair, so that it can be utilized in  N^2  = 36*9 =324
324/18, remainder = 0

C

gmatway · Answer

In simple words we need to make the value of N in a multiple of 36 . But first lets find what can be minimum value of N .

Minimum value has to be such that N SHOULD BE DIVIDED by 108 completely . 108 = 2^2 * 3^2 * 3 so for N to be N^2 it has to have one more 3 SO THAT IT CAN BE DIVIDED by 108 .

so N^2 = 2^2 * 3^2 * 3^2*K = 324K (where K is a no. which maintains the square value of N^2 Hence N becomes 18K (SQUARE ROOT OF 324K) 
NOW N/36 =18K/36 gives us remiander 18 .

NOW this K has to have minimum pairs as 36 have in order to get divided by 36 . So 3^2 * 2k /2^2*3^2 
WE OBSERVED THAT WE NEED ONE MORE 2 so that the N gets completely divided hence N= 3^2* 2^2 will become 36 .
and we get remainder zero . Option 6 cant be as N is minimum 18K.

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ScottTargetTestPrep · Answer

Solution:
 
Recall that the prime factors of a perfect square all have even exponents.

Since 108 = 4 x 27 = 2^2 x 3^3, the smallest value of N^2 is 2^2 x 3^4, and hence the smallest value of N is 2 x 3^2 = 18. Of course, N also could be any multiple of 18. If N = 18k where k is odd, then the remainder when N is divided by 36 is 18. If N = 18k where k is even, then the remainder when N is divided by 36 is 0. So the remainder could be either 0 or 18.

Answer: C

BrushMyQuant · Answer

Given that 108 is a factor of  N^2  and we need to find which of the following could be the remainder when integer N is divided by 36?

108 is a factor of  N^2 
=>  N^2  is a multiple of 108

108 =  2^2 * 3^3 
=>  N^2  is a multiple of  2^2 * 3^3 
Making  2^2 * 3^3  close to the next even power ( 2^2 * 3^4 )

=>  N^2  =  2^2 * 3^4  *  k^2  (where k is an integer)
=> N =  2*3^2 k = 18k

=> N can be
18*0 = 0
18*1 = 18
18*2 = 36
and so on

=> 0/36 remainder is 0
=> 18/36 remainder is 18
=> 36/36 remainder is 0

=> Possible values of remainders are 0 and 18

So,  Answer will be C 
Hope it helps!

Watch the following video to learn the Basics of Factors and Multiples

https://www.youtube.com/watch?v=1rw-WLkavcM

Kinshook · Answer

akurathi12

If 108 is a factor of N^{2} then which of the following could be the remainder when N is divided by 36?

I. o
II. 18
III. 6

A. I only
B. III only
C. I& II only
D. I,II& III
E. None of these

108 = 2^2*3^3

N = 2*3^2*x = 18x

Case 1: N = 36
The remainder when N is divided by 36 = 0

Case 2: N = 18
The remainder when N is divided by 36 = 18

N/36 = 18x/36 = x/2: The remainder when N is divided by 36 is 0 when x is even and the remainder when N is divided by 36 is 18 when x is odd

IMO C
 akurathi12

If 108 is a factor of N^{2} then which of the following could be the remainder when N is divided by 36?

I. o
II. 18
III. 6

A. I only
B. III only
C. I& II only
D. I,II& III
E. None of these

If 108 is a factor of N^{2} then which of the following could be the r

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GMAT Problem Solving (PS) Questions

If 108 is a factor of N^{2} then which of the following could be the r

Prep Toolkit

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