Bunuel wrote:
bibha wrote:
If a, b, and c are positive distinct integers, is (a/b)/c an integer?
1. c=2
2. a = b+c
First of all \(\frac{\frac{a}{b}}{c}=\frac{a}{bc}\). So the question becomes is \(\frac{a}{bc}=integer\) true?
(1) \(c=2\) --> is \(\frac{a}{2b}=integer\). Clearly insufficient. If \(a=1\), then answer is NO, but if \(a=4\) and \(b=1\), then the answer is YES.
(2) \(a=b+c\) --> \(\frac{a}{bc}=\frac{b+c}{bc}=\frac{b}{bc}+\frac{c}{bc}=\frac{1}{c}+\frac{1}{b}\). As \(b\) and \(c\) are distinct integers then \(\frac{1}{c}+\frac{1}{b}\) won't be an integer. Sufficient. (Side note: if \(b\) and \(c\) were not distinct integers then \(\frac{1}{c}+\frac{1}{b}\) could be an integer in the following cases: \(b=c=1\) and \(b=c=2\)).
Answer: B.
Hope it helps.
Thanks for the response. I also chose B but took over 3 minutes because I kept checking to see whether there might be a situation in which the reciprocals of 2 distinct integers can add up to produce an integer. Wondering if there is a mathematical concept / theory that can prove this will never be the case? I know it's unnecessary but will help strengthen our understanding of how reciprocals of positive integers function.
One thing that I can think of:
1.Imagine a number line with 0, 1, 2 and so on. Now both the reciprocals of positive integers a and b will lie between 0 and 1 (inclusive) - no other form is possible given that a and b are positive integers (don't think about distinct integers just yet)
2. Now if the sum of these two fractions (1/a and 1/b) must be an integer there are two possibilities - either a. they sum up to 1 or b. they sum up to 2
-The MAXIMUM value of 1/a or 1/b is 1 - so the maximum sum is 1+1 or 2 (to maximize 1/a we must minimize a, which is a positive integer and 1 is the smallest positive integer)
3. The question now becomes can 1/a+1/b be 1 or 2 with a and be being DIFFERENT integers
a. Sum of 1 - this is only possible at 1/2 and 1/2 as no other fraction can be written of the form 1/integer and summed up to add 1 --> my question to the group is - can this be theorized?
b. Sum of 2 - this is only possible at 1 and 1 as no 2 fractions between 0 and 1 can be summed up to give you 2 (test extreme case 0.999+0.998 = 1.997 which is less than 2)
B produces a definite answer - therefore B
Thoughts welcome!