## If $$a, b$$, and $$c$$ are integers, what is the value of

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### If $$a, b$$, and $$c$$ are integers, what is the value of

by AAPL » Mon May 27, 2019 3:18 am

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If $$a, b$$, and $$c$$ are integers, what is the value of $$a$$?

1) $$2^a+2^b=33$$
2) $$a\cdot c = 5$$

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by ceilidh.erickson » Sat Jun 01, 2019 10:38 am

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If $$a, b$$, and $$c$$ are integers, what is the value of $$a$$?

We're given no information in the question stem except that these variables are integers. So, we have to dive into the statements:

(1) $$2^a+2^b=33$$
Think of combinations of powers of 2 that would add to 33. Since 33 is odd, it must be (odd + even) or (even + odd). The only power of 2 that's odd is $$2^0=1$$ .
$$2^0+2^5=1+32=33$$
We know that one of these values must be 0 and the other 5, but we don't know which is which. Insufficient.

(2) $$a\cdot c = 5$$
If both of these are integers, it must be 1*5 or 5*1. Since we don't know which is which, though, this is insufficient.

(1) and (2) together:
(1) tells us that $$a=0$$ or $$a=5$$, and (2) tells us that $$a=1$$ or $$a=5$$. Using the statements together, it must be the case that $$a=5$$. Sufficient.

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by Ian Stewart » Sat Jun 01, 2019 12:43 pm

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ceilidh.erickson wrote: (2) $$a\cdot c = 5$$
If both of these are integers, it must be 1*5 or 5*1. Since we don't know which is which, though, this is insufficient.
Just because this is so important in so many questions: if ac = 5, and a and c are integers, there are four possibilities, not two: a and c can be 5 and 1, in either order, or they can be -5 and -1, in either order.

Of course, when we combine the two statements, we can discard the negative solutions, but from Statement 2 alone, we have four possible values of a.
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by ceilidh.erickson » Sat Jun 01, 2019 5:06 pm

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Ian Stewart wrote:
ceilidh.erickson wrote: (2) $$a\cdot c = 5$$
If both of these are integers, it must be 1*5 or 5*1. Since we don't know which is which, though, this is insufficient.
Just because this is so important in so many questions: if ac = 5, and a and c are integers, there are four possibilities, not two: a and c can be 5 and 1, in either order, or they can be -5 and -1, in either order.

Of course, when we combine the two statements, we can discard the negative solutions, but from Statement 2 alone, we have four possible values of a.
Ian has an excellent point here! The question doesn't specify non-negative. I think this shows how we all really synthesize before we extrapolate - after reading statement 1, when I read statement 2 I immediately thought "well a=5, but I don't know that from this one alone" and didn't further pick apart what I already knew to be insufficient. But Ian's point is important - we can't assume a non-negative constraint where none is specified.
Ceilidh Erickson
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