If x and y are positive integers and 21x + 23y = z, what is the value

Question

If x and y are positive integers and 21x + 23y = z, what is the value of y? 
 
(1) x = 4
(2) z = 130

pushpitkc · Answer

1. x = 4 
When x=4, 84 + 23y = z
But, without knowing the value for z, we cannot tell what the value for y is(Insufficient)

2. z = 130 
Since z = 130,
21x + 23y = 130
We know that x and y are integers, 
This is possible only when y=2 and x=4  (Sufficient) (Option B)

pushkarajnjadhav · Answer

1) x=4
substitute x=4,
84 + 23y = z
without knowing the value for z, it is impossible to tell the value of y.
 clearly statement 1) is Insufficient.

2) z = 130
substitute z=130
21x + 23y = 130
as x and y are positive integers, only possible solution is x=4 and y=2.
 hence, statement 2) is sufficient.

Answer is Option B.

Kudos if it helps.

BrentGMATPrepNow · Answer

Target question :    What is the value of y?

Given : x and y are positive integers and 21x + 23y = z

Statement 1 : x = 4  
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of x, y and z that satisfy statement 1 (and the given information). Here are two: 
 Case a : x = 4, y = 1 and z = 107, in which case  y = 1 
 Case b : x = 4, y = 2 and z = 130, in which case  y = 2 
Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : z = 130 
NOTE: Since we're told that x and y are POSITIVE INTEGERS, this statement is very limiting. 
To determine whether or not this helps us determine the value of y, we need to check all possible solutions to the equation 21x + 23y = z where z = 130
Since x must be a positive integer, we'll start looking for solutions ( to the equation 21x + 23y = 130) when x = 1
Since (7)(21) = 147, and since 147 is greater than 130, we can see that x cannot equal 7 and x cannot be greater than 7
So, we'll check all values of x from 1 to 6:
 x = 1 . We get: 21(1) + 23y = 130. So, 23y = 109. This equation has no INTEGER solution for y, and since y must be a positive integer, we know that x cannot equal 1
 x = 2 . We get: 21(2) + 23y = 130. So, 23y = 80. This equation has no INTEGER solution for y. So, x cannot equal 2
 x = 3 . We get: 21(3) + 23y = 130. So, 23y = 67. This equation has no INTEGER solution for y. So, x cannot equal 3
 x = 4 . We get: 21(4) + 23y = 130. So, 23y = 46, which means  y = 2 . 
 x = 5 . We get: 21(2) + 23y = 130. So, 23y = 25. This equation has no INTEGER solution for y. So, x cannot equal 5
 x = 6 . We get: 21(2) + 23y = 130. So, 23y = 4. This equation has no INTEGER solution for y. So, x cannot equal 6

We've checked all possible solutions to the equation, and only one of them yielded an INTEGER value for y
So, it must be the case that  y = 2 
Since we can answer the  target question  with certainty, statement 2 is SUFFICIENT

Answer:  C

Cheers,
Brent

BrentGMATPrepNow · Answer

ASIDE: Notice that, with statement 2, we are able to answer the target question EVEN THOUGH we have only 2 equations with 3 variables. 
While it's true that, when the variables can represent REAL NUMBERS, then there are infinitely many solutions to a system of 2 equations with 3 variables, the same cannot be said when we're the variables are limited to POSITIVE INTEGERS (as they are in the above question).

The test-makers LOVE to test this concept.

For more on this, watch the following video:
 https://www.youtube.com/watch?v=9SCrYMCA88A

Cheers,
Brent

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If x and y are positive integers and 21x + 23y = z, what is the value

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If x and y are positive integers and 21x + 23y = z, what is the value

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