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# If a, b, and c are not equal to 0 or 1 and if a^x = b, b^y = c and c^z

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Math Expert
Joined: 02 Sep 2009
Posts: 58428
If a, b, and c are not equal to 0 or 1 and if a^x = b, b^y = c and c^z  [#permalink]

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Updated on: 23 Jul 2019, 10:18
00:00

Difficulty:

25% (medium)

Question Stats:

78% (01:40) correct 22% (01:31) wrong based on 50 sessions

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If $$a$$, $$b$$, and $$c$$ are not equal to 0 or 1 and if $$a^x=b$$, $$b^y=c$$, and $$c^z = a$$, then $$xyz=$$

(A) 0

(B) 1

(C) 2

(D) a

(E) abc

Source: Nova GMAT
Difficulty Level: 600

_________________

Originally posted by Bunuel on 07 Apr 2019, 21:39.
Last edited by SajjadAhmad on 23 Jul 2019, 10:18, edited 1 time in total.
Intern
Joined: 09 Apr 2018
Posts: 32
Re: If a, b, and c are not equal to 0 or 1 and if a^x = b, b^y = c and c^z  [#permalink]

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07 Apr 2019, 21:56
Since a,b and c can't be 0 or 1, x, y and z have to be 1.
Hence xyz=1

Alternately,

Since we have a^x=b & b^y=c, (a^x)^y=c
a^xy=c
Again,
c^z=a And hence (a^xy)^z=a
a^xyz=a
Intern
Joined: 24 Jun 2018
Posts: 35
Re: If a, b, and c are not equal to 0 or 1 and if a^x = b, b^y = c and c^z  [#permalink]

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08 Apr 2019, 11:42
Simple substitution can give us the answer:

Given:
a^x = b
b^y=c
c^z=a

Now,
b^y=c
=> (a^x)^y=c
=> a^xy=c ------> (1)

Now,
c^z=a
From 1
(a^xy)^z=a
Hence, a^xyz=a

Since a is not equal to 0 or 1, for both sides to be equal

xyz must be equal to 1
hence xyz = 1
Senior Manager
Joined: 12 Sep 2017
Posts: 302
Re: If a, b, and c are not equal to 0 or 1 and if a^x = b, b^y = c and c^z  [#permalink]

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09 Apr 2019, 18:01
1
I did it this way, just keep adding the exponents, given:

$$a^x = b$$
$$b^y = c$$
$$c^z = a$$

Take the first one and add ^y to both sides.

$$a^x^y = b^y = c$$ Now add the ^z so the whole term can be equal to a.

$$a^x^y^z = b^y^z = c^y^z = a$$

so

$$a^x^y^z = a^1$$

$$xyz = 1$$

B
Re: If a, b, and c are not equal to 0 or 1 and if a^x = b, b^y = c and c^z   [#permalink] 09 Apr 2019, 18:01
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