If a and b are both integers, how many possible solutions are there to

possible values -7<a<7
for all even 'a's there is an integer value for b.
thus the solution points are 0, +|- 2,4,6.

hence 7. D

Hi All,

In this question, you can use Number properties to your advantage (as some of the others posters have pointed out). If you got "stuck" on this question, then you should note that the answer choices are relatively small and the math is basic arithmetic. This means that you can use "brute force" to get to the answer. You can easily determine the solutions by taking some notes and doing some basic math.

We're told:
1) A and B are both INTEGERS
2) 3A + 2B = 50
3) 7 > |-A|

We're asked for the total number of POSSIBLE solutions to the equation. From the answers, we know that there are at least 4 and at most 8. Let's find them....

Keep things simple at first...and take notes so that you can make deductions...
IF....
A = 0, then B = 25

IF....
A = 1, then B = 23.5
This is NOT a possible solution (since B is NOT an integer).
This tells us that A CANNOT be ODD, since that causes B to become a non-integer.
From here on, we WON'T WASTE TIME testing ODD numbers for A....

IF....
A = 2, then B = 22

IF....
A = 4, then B = 19
Notice the pattern emerging: when A increases by 2, B decreases by 3.....

IF....
A = 6, then B = 16

We're not allowed to go any higher (the inequality limits us here). So far, we have 4 answers. The question NEVER stated that A and B couldn't be negative though....and that 'absolute value' IS there for a REASON.....

IF....
A = -2, then B = 28

IF...
A = -4, then B = 31

IF...
A = -6, then B = 34

Now, we're not allowed to go any lower (again, the inequality limits us here). We now have 3 additional answers.

4 + 3 = 7 total answers.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

i wonder what the stats were if one of the answers was "13".

i must admit that i would mark that one...

My take is Option D. Used a simple approach.

Going by 2nd equation, possible values of a are : -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6

1st equation: 50 = 3a + 2b
=> b = (50-3a) / 2
As per the question stem, both a and b are integers. Hence 50 - 3a should be even. Hence, a should be even - This gives us 6 values. Now remember 50 - 3a = even, if a is 0.
So we have 7 values for a (-6, -4, -2, 0, 2, 4, 6). Total 7 solutions.

Thanks,
Chanakya

Hit kudos if you like the explanation!!

Its very simple.
Before solve this solution lets discus 2 things:-
1) when we write |-x| this means the same as |x|.
2) Mod is very easy concept if you solve mod question by considering as a distance. when a mod is written as |x-(a)| = b, this means the distance from point 'a' (both side left and right of 'a' on number line) is b. |x-(a)| < b means the distance is between the two extreme distance(left and right side of 'a' on number line, considering the max distance is 'b' from 'a' - as per this scenario.....hence the value of 'a' must be between these two extremes. |x-(a)| > b means the distance is greater than the distance of 'b'..i.e the value of a could be anywhere more than 'b'.

Now come to the question. First its given|-a| < 7 ==> |a| < 7 ===> |a-0| < 7==> the distance from zero is less than 7. So the point will be -7 and 7, as distance from 0 to (-7) is 7 and distance from 0 to 7 is 7. Thus, the value of a will lie in between -7 to 7....i.e the value of a (integer given in ques) would be -6, -5, -4,-3,-2,-1,0,1,2,3,4,5,6.

Now, lets move to equation 3a + 2b = 50 ==> b = 25 -(3/2)a. According to question, b is an integer, hence to make b integer a must be divisible by 2. Now remove the value which can not be divisible by 2 from the possible values of a. It will remain with -6,-4,-2,0,2,4,6...i.e total = 7...hence ans is D

+1 Kudos if it helped you.

damn..I did an error an wrote 7<|-a|
and this messed up everything

Hi mvictor,

It's perfectly fine to make mistakes during practice, but there's a bigger 'takeaway' from all of this. Silly/little mistakes tend to 'kill' Test Takers, and regardless of how far you might be from your score goal, they're likely hurting your performance. You'd be amazed how many of the questions you face on Test Day are 'gettable'... as long as you take the necessary notes, stay organized and do the work ON THE PAD.

GMAT assassins aren't born, they're made,
Rich

Given,

|-a|<7

-7<-a<7

7>a>-7

-7<a<7.

a = -6, - 5, -4, -3, -2, -1, 0, 1 ,2, 3, 4, 5, 6,

3a + 2b = 50

2b = 50 - 3a

b = 50 - 3a / 2.

Now , both a and b are integers thus 3a has to be even. In order to do so, a has to be even.

There are 7 even values of a. For each even value of a , b will have an integer value.

Thus , correct answer is D.

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