Question 29
Positive integers $$a, b, c$$ satisfy $$\frac{ab}{a-b} = c$$. What is the largest possible value of $$a+b+c$$ not exceeding 99?
Correct Answer: 99
Solution
We need $$a > b$$ and $$(a-b) \mid ab$$. Taking $$a = 27$$ and $$b = 18$$ gives $$a - b = 9$$ and $$c = \frac{27 \times 18}{9} = 54$$, so $$a+b+c = 27+18+54 = 99$$. Since the bound 99 is attained, the largest possible value is 99.
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