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# If a, b and c are different real numbers such that a+1/b = b+1/c = c+1

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VP
Joined: 19 Oct 2018
Posts: 1174
Location: India
If a, b and c are different real numbers such that a+1/b = b+1/c = c+1  [#permalink]

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19 Jun 2019, 17:53
2
00:00

Difficulty:

45% (medium)

Question Stats:

63% (02:08) correct 38% (02:27) wrong based on 40 sessions

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If a, b and c are different real numbers such that $$a+\frac{1}{b}$$ = $$b+\frac{1}{c}$$ = $$c+\frac{1}{a}$$ , which of the following is a possible value of the a*b*c?

A. 1
B. 2
C. 3
D. 4
E. 8
VP
Joined: 20 Jul 2017
Posts: 1147
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
If a, b and c are different real numbers such that a+1/b = b+1/c = c+1  [#permalink]

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Updated on: 19 Jun 2019, 18:58
2
1
nick1816 wrote:
If a, b and c are different real numbers such that $$a+\frac{1}{b}$$ = $$b+\frac{1}{c}$$ = $$c+\frac{1}{a}$$ , which of the following is a possible value of the a*b*c?

A. 1
B. 2
C. 3
D. 4
E. 8

**Edited

OK! Not sure a GMAT question or not? Definitely took me > 2 min

Given,
$$a+\frac{1}{b}$$ = $$b+\frac{1}{c}$$
--> $$a - b$$ = $$\frac{1}{c}$$ - $$\frac{1}{b}$$ = $$\frac{(b - c)}{bc}$$ ....... (1)

$$b+\frac{1}{c}$$ = $$c+\frac{1}{a}$$
--> $$b - c$$ = $$\frac{1}{a}$$ - $$\frac{1}{c}$$ = $$\frac{(c - a)}{ac}$$ ....... (2)

$$a+\frac{1}{b}$$ = $$c+\frac{1}{a}$$
--> $$a - c$$ = $$\frac{1}{a}$$ - $$\frac{1}{b}$$ = $$\frac{(b - a)}{ab}$$ ....... (3)

Multiply (1), (2) & (3)
--> $$(a - b)*(b - c)*(a - c)$$ = ($$\frac{(b - c)}{bc}$$)*($$\frac{(c - a)}{ac}$$)*($$\frac{(b - a)}{ab}$$)
--> $$(a - b)*(b - c)*(a - c)$$ = $$\frac{(b - c)*(c - a)*(b - a)}{(abc)^2}$$

The above equation is only possible when (abc)^2 = 1
--> a*b*c can be 1 or -1

So, possible value = 1

IMO Option A

Pls Hit kudos if you like the solution

Originally posted by Dillesh4096 on 19 Jun 2019, 18:18.
Last edited by Dillesh4096 on 19 Jun 2019, 18:58, edited 1 time in total.
VP
Joined: 19 Oct 2018
Posts: 1174
Location: India
Re: If a, b and c are different real numbers such that a+1/b = b+1/c = c+1  [#permalink]

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19 Jun 2019, 18:27
a, b and c are different, though your answer might be correct.

Dillesh4096 wrote:
nick1816 wrote:
If a, b and c are different real numbers such that $$a+\frac{1}{b}$$ = $$b+\frac{1}{c}$$ = $$c+\frac{1}{a}$$ , which of the following is a possible value of the a*b*c?

A. 1
B. 2
C. 3
D. 4
E. 8

We can clearly see that if each of a, b & c is 1, the given equation is satisfied

So, definitely a*b*c can be 1

IMO Option A

Pls Hit kudos if you like the solution

Posted from my mobile device
VP
Joined: 20 Jul 2017
Posts: 1147
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
If a, b and c are different real numbers such that a+1/b = b+1/c = c+1  [#permalink]

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19 Jun 2019, 18:30
nick1816 wrote:
a, b and c are different, though your answer might be correct.

Dillesh4096 wrote:
nick1816 wrote:
If a, b and c are different real numbers such that $$a+\frac{1}{b}$$ = $$b+\frac{1}{c}$$ = $$c+\frac{1}{a}$$ , which of the following is a possible value of the a*b*c?

A. 1
B. 2
C. 3
D. 4
E. 8

We can clearly see that if each of a, b & c is 1, the given equation is satisfied

So, definitely a*b*c can be 1

IMO Option A

Pls Hit kudos if you like the solution

Posted from my mobile device

Oh Damn! Lemme redo
Senior Manager
Joined: 12 Sep 2017
Posts: 308
Re: If a, b and c are different real numbers such that a+1/b = b+1/c = c+1  [#permalink]

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25 Jun 2019, 19:59
I think it must have something related with reciprocals and tahts why is 1, but idk how to solve it.
Re: If a, b and c are different real numbers such that a+1/b = b+1/c = c+1   [#permalink] 25 Jun 2019, 19:59
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