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# If 3x4y = 177,147 and x – y = 11, then x =?

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Math Expert
Joined: 02 Sep 2009
Posts: 60480
If 3x4y = 177,147 and x – y = 11, then x =?  [#permalink]

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29 Feb 2016, 12:49
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Difficulty:

35% (medium)

Question Stats:

73% (02:02) correct 27% (02:16) wrong based on 143 sessions

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If 3^x*4^y = 177,147 and x – y = 11, then x =?

A) Undefined
B) 0
C) 11
D) 177,136
E) 177,158

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Joined: 12 Sep 2015
Posts: 4214
Re: If 3x4y = 177,147 and x – y = 11, then x =?  [#permalink]

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29 Feb 2016, 12:55
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3
Bunuel wrote:
If (3^x)(4^y) = 177,147 and x – y = 11, then x =?

A) Undefined
B) 0
C) 11
D) 177,136
E) 177,158

Great question, Brunuel!

Notice that, since 177,147 is ODD, it is not a multiple of 4.
This means that y MUST equal zero.
If y = 0 AND we know that x - y = 11, we can conclude that x = 11

Cheers,
Brent
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Re: If 3x4y = 177,147 and x – y = 11, then x =?  [#permalink]

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29 Feb 2016, 13:35
Bunuel wrote:
If 3^x*4^y = 177,147 and x – y = 11, then x =?

A) Undefined
B) 0
C) 11
D) 177,136
E) 177,158

177147 does not contain any power of 2, that means y should be 0 => x=11
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Re: If 3x4y = 177,147 and x – y = 11, then x =?  [#permalink]

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03 Mar 2016, 03:08
Bunuel wrote:
If 3^x*4^y = 177,147 and x – y = 11, then x =?

A) Undefined
B) 0
C) 11
D) 177,136
E) 177,158

This is a question which needs your keen observation rather than calculation powers.
The first thought that comes into mind after looking at such questions is to factorize the number.

When we try to do that, we see that 177,147 is odd, therefore 4^y = 1 or y = 0
We have x - y = 11
Therefore x = 11

Option C
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Re: If 3x4y = 177,147 and x – y = 11, then x =?  [#permalink]

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03 Mar 2016, 18:33
Good question. Y must be zero since 177,147 is without an even. This leaves X = 11 because the two others cannot be the powers of three since the 3^x*4^y = 177147 must be true. They would be too big. The number doesn't divide into 3 eleven times either therefore the answer must be irrational, so IMO A. undefined.
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Re: If 3x4y = 177,147 and x – y = 11, then x =?  [#permalink]

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03 Mar 2016, 18:37
Aw dang, answer fail. I used the calculator shortly after answering the question. I wouldn't let myself not put my answer up first before doing so haha. The thing is, how could deduce that three divides into that number? I couldn't think of any quick tricks, and unless you have a trick, it's 50/50 (which isn't bad odds on this test though.. but still)
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Posts: 351
Re: If 3x4y = 177,147 and x – y = 11, then x =?  [#permalink]

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11 Dec 2019, 22:07
If the base is odd, the final result will always be odd regardless of the exponent to which it is raised. Similarly, if the base is even, the final result will always be even regardless of the nature of the exponent.

This is based on the fact that exponentiation is repeated multiplication and repeatedly multiplying odd numbers will give you an odd result and vice-versa.

Considering the above, $$3^x$$ * $$4^y$$ = 177,147 can happen only if $$4^y$$ is odd. If $$4^y$$ was even, the product can never be 177,147 (an odd number). The only way in which $$4^y$$ can be odd is when y=0. Remember, any number (other than ZERO itself) raised to the power of 0 equals 1.

Substituting the value of y=0 in the equation x-y = 11, we get x = 11. It’s not just a coincidence then that $$3^{11}$$ = 177,147 ?

An alternative approach to solve this question could be to use the cyclicity of units digits of the power of 3, but only after figuring out that $$4^y$$ has to be odd. The cyclicity of unit digits of 3 is 3,9,7 and 1. Since the number we have is a number ending with 7, the power should be of the form 4k+3 and this also points us towards the fact that x=11.

The correct answer option is C.

Hope that helps!
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Re: If 3x4y = 177,147 and x – y = 11, then x =?  [#permalink]

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06 Jan 2020, 05:31
Bunuel wrote:
If 3^x*4^y = 177,147 and x – y = 11, then x =?

A) Undefined
B) 0
C) 11
D) 177,136
E) 177,158

You can take a good guess that there will be at least 3^2 in the term.

So keep on dividing by 9 and then by 3 if needed.

177,147 divided by 9 = 19683
19683 divided by 9 = 2187
Keep on dividing by 9 : 243,27,3
3 is divisible by 3.

So we got 5 9s and 1 3s. That's 11 3s. That means y is 0 and x is 11.

Option C
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Re: If 3x4y = 177,147 and x – y = 11, then x =?   [#permalink] 06 Jan 2020, 05:31
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