For how many positive integer values of N, less than 40 does the equat

Question

For how many positive integer values of N, less than 40 does the equation 3a – Nb = 5, have no integer solution?

A. 11
B. 12
C. 13
D. 14
E. 15

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nick1816 · Accepted Answer

Tough one  Bunuel . Giving it try tho.

3a - Nb = 5

3 cases are possible

1. N= 3k

3a-3kb =5
3(a-kb)=5
a-kb = 5/3

since a, k and b are integers, a-kb can't be a fraction. hence, N can never be a multiple of 3.

2. N=3k+1

3a-3kb-b=5
3(a-kb)-b=5
3(a-kb)=5+b

if b is in the form of 3x+1(x is an integer), there is always at least 1 integral solution possible

3. N=3k+2

3a-3kb-2b=5
3(a-kb)-2b=5
3(a-kb)=5+2b

if b is in the form of 3x+2 (x is an integer), there is always at least 1 integral solution possible.

Hence, there is no integral solution possible for equation 3a – Nb = 5 when N is multiple of 3.

There ate 13 multiples of 3 in the range from 0-40(both exclusive)

TestPrepUnlimited · Answer

Arrange the equation to have  3a - 5 = Nb . The left side is a multiple of 3 minus 5, so the left side CANNOT be a multiple of 3. Then if the right side was a multiple of 3, we would certainly not have integer solutions.

Therefore we pick out all multiples of three's below 40, which is multiples of three from 3 to 39 inclusive and that is 13 integers.

Ans: C

The reader might question what if N isn't a multiple of 3. However the answers only go from 11 to 15, so we can assume other N's (N = 3i + 1 and N = 3i + 2) will have a viable integer solution due to the small range from the answer choices. For a rigorous proof see the post from above.

ShreyasJavahar · Answer

Hi Nick, could you please explain how you arrived at the 3 cases that you've mentioned?

Kinshook · Answer

Asked: For how many positive integer values of N, less than 40 does the equation 3a – Nb = 5, have no integer solution?

If N is a multiple of 3, Left side of the equation will become multiple of 3 and right side =5 is not a multiple of 3
Therefore, N can not be a multiple of 3.

N = {3,6,9.....,39}
number of values of N = (39-3)/3 + 1 = 13

IMO C

Kinshook · Answer

Hi  ShreyasJavahar

Any integer can be written in the forms 3k, 3k+1 or 3k+2 since if you divide any integer by 3, remainders you can get will be 0,1 or 2.

DrAnkita91 · Answer

In some GMAT questions, positive integers include zero as well.. 
I am confused.. Please throw some light on it
whether to consider zero as a positive integer

Bunuel · Answer

Zero is an even integer that is  neither positive nor negative . No proper GMAT question would suggest otherwise.

ZERO:

1. Zero is an  INTEGER .

2. Zero is an  EVEN  integer. An even number is an   integer   that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.

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

4. Zero is  divisible by EVERY integer except 0 itself  ( \frac{x}{0} = 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 .

nipun7399 · Answer

Given -> 3a - Nb = 5,
N = +ve int <40 --> N ={1,2,3............39}

Since question says that solution shouldn't be an intiger so to make RHS a fraction let's assume N = 3k, where k = {1,2,3.....} [+ve int]

Now, 3a - Nb =5
3a - 3kb = 5
3(a - kb) = 5
a - kb = 5/3 = non int

Now N= 3k < 40 & k = {1,2,3.....}
therefore N = {3,6,9.......39}

and hence for 13 values of N, equation cannot have an intiger solution

For how many positive integer values of N, less than 40 does the equat

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For how many positive integer values of N, less than 40 does the equat

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