monsoon1 wrote:
If a, b, and c are positive integers such that 1/a + 1/b = 1/c, what is the value of c?
(1) b ≤ 4
(2) ab ≤ 15
Source : Manhattan
Advanced Quant Question No. 3
OFFICIAL SOLUTION
Since a, b, and c are positive integers, 1/a, 1/b, and 1/c are each less than or equal to 1. Also, 1/a and 1/b must both be less than 1/c, implying that a and b must both be greater than c. Furthermore, either 1/a or 1/b must be no less than ½ of 1/c, because if both fractions are less than ½ of 1/c, the sum will be less than 1/c , which implies that either a or b must be less than or equal to 2c.
These implications, along with the integer constraints and the given equation, greatly reduce the number of possible values for a, b, and c. We should make a comprehensive list for the first few c values. A good approach is to work backwards from the target value of c (= 1, 2, 3, etc.) and try to find integer values of a and b that fit the equation. There are only a few possibilities in each case. One pair that always works is making both a and b equal to 2c. Also, if we make a equal to c + 1, then there is always an integer value for b (which winds up equaling ac or c(c + 1), as we can show by a little algebra).
This question can be rephrased to “Which (a, b) pairs listed above are valid, and what is the resulting value of c?”
(1) INSUFFICIENT: If b ≤ 4, then valid (a, b) pairs are (2, 2) and (6, 3) and (4, 4) and (12, 4). This implies that c could be 1, 2, or 3.
(2) SUFFICIENT: If ab ≤ 15, then the only valid (a, b) pair is (2, 2) and c must be 1.
By the way, fractions of the form 1/integer are known as Egyptian fractions, because they were used first in ancient Egypt.
The correct answer is B.
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