A function "a#b" is defined for integers a and b such that a#b = a+b

Question

A function "a#b" is defined for integers a and b such that a#b = a+b if both a and b are odd, and a#b = (a*b)/2 otherwise. For integers x and y, what is x#y?

(1) |y| < 1.

(2) x = 14k, where 'k' is a non negative integer.

DavidTutorexamPAL · Answer

Instead of trying to figure out what kind of information we need, we'll try simple examples.
This is an Alternative approach.

(1) Since y is an integer, it must equal 0. Then x#y = (x*0)/2 = 0
Sufficient.

(2) So x is definitely even meaning that x#y = xy/2. But since we can choose many different values for both x and y, this is insufficient.
Insufficient.

(A) is our answer.

chetan2u · Answer

A function "a#b" is defined for integers a and b such that a#b = a+b if both a and b are odd, and a#b = (a*b)/2 otherwise. For integers x and y, what is x#y?

(1) |y| < 1.
This means distance of y from 0 is within 1 units so -1<0<1
even by opening modulus..
a) y>0......y<1
b) y<0......-y<1......y>-1
so -1<y<1
so integer value of y possible is 0
and as y is even \(x#y=\frac{x*y}{2}=\frac{x*0}{2}=0\)
sufficient

(2) x = 14k, where 'k' is a non negative integer.
nothing about y
insuff

A

sapog · Answer

In the explanation it is said, that

"(2) So x is definitely even meaning that x#y = xy/2. But since we can choose many different values for both x and y, this is insufficient.
Insufficient.

(A) is our answer."

So why do we need to know what Y is equal to, if it is mentioned in the task, that if one of the values: a or b, is not odd, then we do the following  a#b = (a*b)/2
My point is exactly that a#b can be odd-odd, odd-even, even-even, even-odd and for those cases where  a and b are not odd  we use this function a#b = (a*b)/2

egmat · Answer

Hi sapog,

You're actually  completely right  about the formula, and that's the good news. Since Statement  (2)  makes x even (x =  14 k =  0 ,  14 ,  28 , ...), the "both odd" case is impossible, so the rule that applies is definitely  x#y = (x·y)/2 . No argument there.

But here's the thing the question is really asking: it wants the  value  of x#y - an actual number. Knowing  which formula  to use isn't the same as knowing  what it evaluates to .

Look at that formula again:  (x·y)/2 . It still has a  y  sitting inside it. So even after you've locked in the right formula, the answer moves around as y changes.

Two quick cases (both obey Statement 2): 
- k =  1 , y =  1  - x =  14  - x#y = ( 14 · 1 )/ 2  =  7 
- k =  1 , y =  2  - x =  14  - x#y = ( 14 · 2 )/ 2  =  14

Same statement, same formula - but  7  in one case and  14  in the other. Two different values means Statement  (2)  is  not sufficient .

Compare that with Statement  (1) : there y is pinned to  0 , so  (x·0)/2 = 0  no matter what x is - one definite value,  sufficient .

The takeaway:  picking the correct branch of the function is only step one. In a  value  DS question, you're not done until the expression collapses to a single number. If a variable in the chosen formula is still free to change, the answer changes with it - and that's exactly why  (2)  falls short and the answer is  A .

Answer: A

A function "a#b" is defined for integers a and b such that a#b = a+b

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GMAT Data Sufficiency (DS) Questions

A function "a#b" is defined for integers a and b such that a#b = a+b

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