If $a, b,$ and $c$ are positive integers such that $\dfrac1{a} + \dfrac1{b} = \dfrac1{c},$ what is the value of

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Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pm; Thanked: 6 times; Followed by:5 members

If $a, b,$ and $c$ are positive integers such that $\dfrac1{a} + \dfrac1{b} = \dfrac1{c},$ what is the value of

by Vincen » Fri Jun 19, 2020 2:45 am

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If $a, b,$ and $c$ are positive integers such that $\dfrac1{a} + \dfrac1{b} = \dfrac1{c},$ what is the value of $c?$

(1) $b\le 4$
(2) $ab \le 15$

[spoiler]OA=B[/spoiler]

Source: Manhattan GMAT

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Source: Beat The GMAT — Data Sufficiency | Fri Jun 19, 2020 2:45 am

deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pm; Followed by:5 members

Re: If $a, b,$ and $c$ are positive integers such that $\dfrac1{a} + \dfrac1{b} = \dfrac1{c},$ what is the value o

by deloitte247 » Sat Jun 20, 2020 3:09 am

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$$\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$$
$$\frac{b+a}{ab}=\frac{1}{c}$$
$$\frac{c\left(b+a\right)}{\left(b+a\right)}=\frac{ab}{\left(b+a\right)}$$
$$c=\frac{ab}{b+a}$$
$$Given\ that\ c\ is\ a\ positive\ integer\frac{ab}{b+a}must\ be\ divisible\ without\ remainder$$

Target question => What is the value of c?

$$Statement\ 1\ =>\ b\le4$$
The value of a is not given, it can be any value between 1 and infinity. So, the information provided cannot give a definite answer. Statement 1 is NOT SUFFICIENT

$$Statement\ 2\ =>\ ab\le15$$
There are numerous combination of numbers in which their product is < 15 but remember that c is a positive integer and ab/(b+a) is expected to be divisible without remainder
This is only possible when a = 2 and b = 2
$$c=\frac{2\cdot2}{2+2}=\frac{4}{4}=1$$
Statement 2 alone is SUFFICIENT

Answer = B