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# If a, b, and c are positive integers such that 1/a + 1/b = 1

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Intern
Joined: 12 Jan 2017
Posts: 10
Location: United States
GMAT 1: 740 Q50 V40
Re: If a, b, and c are positive integers such that 1/a + 1/b = 1  [#permalink]

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08 Mar 2017, 02:58
VeritasPrepKarishma wrote:
sharkr wrote:
VeritasPrepKarishma wrote:
Basically, you have to look for values such that when the numerators add up, the sum is divisible by the denominator. a and b cannot be 1 since we need the sum to be less than 1.

What are the numerators, the sum, and the denominator referring to?

Thanks in advance.

1/a + 1/b = 1/c

You have to look for values of a and b (denominators) such that when you add 1/a and 1/b, the numerator you get is divisible by the denominator of the fraction (which you get after adding)

e.g. if a = 3, b = 6

1/a + 1/b = 1/3 + 1/6 = (2 + 1)/6 = 3/6
The numerator is 3 and the denominator is 6. The numerator is divisible by the denominator such that you get 1 in the numerator after cancelling.

1/a + 1/b = 1/3 + 1/6 = 1/2

When you get 1 in the numerator, the denominator of the resulting fraction is the value of c (i.e. 2 above)

Thanks for the reply.
For clarification, don't you mean the denominator (ie. 6) is divisible by the numerator (ie. 3)?
3 / 6 = 0*6 + 3 whereas 6/3 = 3*2 (no remainder).
Intern
Joined: 09 May 2016
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Re: If a, b, and c are positive integers such that 1/a + 1/b = 1  [#permalink]

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05 Apr 2017, 14:49
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

Is there any algebric approach to be sure that B is sufficient
Manager
Joined: 10 Dec 2011
Posts: 78
Location: India
Concentration: Finance, Economics
GMAT Date: 09-28-2012
WE: Accounting (Manufacturing)
Re: If a, b, and c are positive integers such that 1/a + 1/b = 1  [#permalink]

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04 Jan 2018, 09:03
mau5 wrote:

From F.S 1, we know that for a to be positive, c<b. The given equation is valid for b=2,c=1 and also for b=3,c=2. Insufficient.

Now, back to your question.

We know that$$\frac{a+b}{2}\geq{\sqrt{ab}}$$

Also, from the question stem, we know that $$\frac{a+b}{ab} =\frac{1}{c}$$

Thus, $$(a+b) = \frac{ab}{c}$$. Replacing this in the first equation, we get $$\frac{ab}{2c}\geq{\sqrt{ab}}$$

Or,$$c\leq{\frac{\sqrt{ab}}{2}} \to c\leq{\frac{\sqrt{15}}{2}} \to c<{2}$$. Thus, the only positive integer less than 2 is One and thus c=1.Sufficient.

Whoa! That's some really interesting point you are making. Never heard of this... that the average of two average of two numbers is always greather than the square root of the product of two numbers! Does this property hold always?
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Re: If a, b, and c are positive integers such that 1/a + 1/b = 1  [#permalink]

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17 Jan 2020, 08:13
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Re: If a, b, and c are positive integers such that 1/a + 1/b = 1   [#permalink] 17 Jan 2020, 08:13

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# If a, b, and c are positive integers such that 1/a + 1/b = 1

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