M06-07

Question

If  2^{98} = 256*m + n , where  m  and  n  are integers and  0 \le n \le 4 , what is the value of  n ?

A. 0
B. 1
C. 2
D. 3
E. 4

Bunuel · Accepted Answer

Official Solution:

If  2^{98} = 256*m + n , where  m  and  n  are integers and  0 \le n \le 4 , what is the value of  n ?

A. 0
B. 1
C. 2
D. 3
E. 4

Given: 
  
 2^{98} = 2^8 * m + n . 
  
Divide both sides by  2^8 : 
  
 2^{90} = m + \frac{n}{2^8} . 
  
Since both  2^{90}  and  m  are integers, then  \frac{n}{2^8}  must also be an integer. Considering the constraint  0 \le n \le 4 , this is only possible when  n = 0 .

Answer: A

Bunuel · Answer

If n = 0, then n/2^8 = 0/2^8 = 0, which is an integer.

ZERO:

1. Zero is an  INTEGER .

2. Zero is an  EVEN  integer.

3. Zero is  neither positive nor negative  (the only one of this kind)

4. Zero is  divisible by EVERY integer except 0 itself  ( \frac{0}{x} = 0 , so 0 is a divisible by every number, x).

5. Zero is a  multiple of EVERY integer  ( x*0 = 0 , so 0 is a multiple of any number, x)

6. Zero is  NOT a prime number  (neither is 1 by the way; the smallest prime number is 2).

7. Division by zero is NOT allowed:  anything/0 is undefined .

8. Any non-zero number  to the power of 0 equals 1  ( x^0 = 1 )

9.  0^0  case is  NOT tested on the GMAT .

10. If the exponent n is positive (n > 0),  0^n = 0 .

11. If the exponent n is negative (n < 0),  0^n  is undefined, because  0^{negative}=0^n=\frac{1}{0^{(-n)}} = \frac{1}{0} , which is undefined.  You CANNOT take 0 to the negative power .

12.  0! = 1! = 1 .

rishi02 · Answer

I think this is a  high-quality  question and I agree with explanation.

pranjal123 · Answer

Another way to approach this problem in terms of ODD and EVEN.

Given :  2^{98}=256L+N

==>  2^{98}=2^8L+N  .Now , Left hand expression is EVEN and we know E+E or O+O only gives output as EVEN But since L is multiplied by 2^8 so ODD+ODD case is not possible.

Now , N can take only even values i.e. 0,2,4 
If we Put N=2 then one 2 from N and L will cancel out from Right and Left side leaving below expression :

2^{97}=2^7L+1

But then Right side expression will give ODD value which is incorrect.Same is the case with N= 4 .

So only possible value is 0 .  Ans A

Madhavi1990 · Answer

Its great question and I'd like to practice a few more of these!

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

vedha0 · Answer

pretty easy but worded trickily. nice question.

Silcodio · Answer

I did not quite understand the solution. There is no explanation on why N needs to be 0

Bunuel · Answer

Here is an explanation:

Since both  2^{90}  and  m  are integers, then   \frac{n}{2^8}  must also be an integer .  Considering the constraint  0 \le n \le 4 , this is only possible when  n = 0  .

So, n can only be 0, 1, 2, 3, or 4. Only if n is 0 do we get an integer value for n/2^8.

Ejvincent · Answer