The number of ordered pairs of positive integers (x, y) satisfying the

Question

The number of ordered pairs of positive integers (x, y) satisfying the equation 7x + 3y = 123 is:

A. 3
B. 5
C. 12
D. 13
E. Infinite

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GMATWhizTeam · Accepted Answer

Solution:  
 • x and y are positive integers.
•  7x + 3y = 123 
 o  3y = (123-7x)  
• We need to find the number of positive values of  x  for which  123 -7x  is a multiple of  3 .
•  123 – 7x  is a multiple of 3 when x is also a multiple of 3.
 o  123-7x > 0 
o  123/7 > x 
o  17.57>x  
• Now the possible values of  x  are  3, 6, 9, 12, 15 , i.e. there are  5  possible values of  x  and for each  x  there will be a unique  y .
• There are  5  ordered pairs of  (x, y) , for which the given equation is true. 
Hence, the correct answer is  Option B.

mdsaddamforgmat · Answer

straight line/Linear equations have infinite ordered pair

please add/correct your views

GMATinsight · Answer

7x + 3y = 123

123 is divisible by 3 i.e. 7x should also be divisible by 3
so first pair of (x, y) = (3, 34)

Now, next value of x will differ by co-efficient of y which is 3 and next value of y will differ by co-efficient of x which is 7

Now, if x increased by 3 then y will decrease by 7

i.e. 7*4 = 28 is maximum that can be subtracted from 34 to keep it positive

So total Solutions = 1+4 = 5

Answer: Option B

GMATinsight · Answer

Question mentioned positive integer solution. I hope this helps!!!

mdsaddamforgmat · Answer

Oh... Silly mistake

Posted from my mobile device

TheUnicorn · Answer

N0te that 123 is divisible by 3. so Any 7x where x> 0, x is multiple of 3, and 7x < 123 will satisfy the given equation. Max value of such 7x = 105 where x =5.

Jawad001 · Answer

The number of ordered pairs of positive integers (x, y) satisfying the equation 7x + 3y = 123 is:

A. 3
B. 5
C. 12
D. 13
E. Infinite

Solution: 7x + 3y = 123
3y = 123-7x
Y = 41-7x/3 ……………….(1)
To be integer the value of y, 41-7x/3 must be integer. So the value of x must be multiple of 3.
When, x = 3,6,9,12,15, then, y = positive integer. 
When x = 18, y = 41 – 7/3(18) 
 • y = 41 – 42 = -1, Not acceptable.
Therefore, there are 5 ordered pair of positive integers.
Answer: B

Kinshook · Answer

Asked: The number of ordered pairs of positive integers (x, y) satisfying the equation 7x + 3y = 123 is:

7x + 3y = 123
x = (123-3y)/7

(x,y) = (9,20) is a solution
x will change by 3 and y will change in opposite sign by 7 for each solution

(x,y) = {(3,34), (6,27), (9,20), (12,13),(15,6)}

IMO B

Kinshook · Answer

But it is mentioned that solution is valid for only positive integers.

WIZARD1234 · Answer

Correct but Qs has specified for positive integers so finite solution sets

Raji11 · Answer

zero is positive integer?

Bunuel · Answer

No. Zero is neither positive nor negative (the only one of this kind).

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 .

Hope it helps.

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