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If x and y are integers and 7x−7x−2=3∗14y, what is the value of x?

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If x and y are integers and 7x−7x−2=3∗14y, what is the value of x?  [#permalink]

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New post 08 Mar 2017, 00:38
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If x and y are integers and \(7^x−7^{x−2}=3∗14^y\), what is the value of x?

A. 2
B. 3
C. 5
D. 6
E. 8

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If x and y are integers and 7x−7x−2=3∗14y, what is the value of x?  [#permalink]

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New post 08 Mar 2017, 01:05
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ziyuen wrote:
If x and y are integers and \(7^x−7^{x−2}=3∗14^y\), what is the value of x?

A. 2
B. 3
C. 5
D. 6
E. 8



7^x-7^(x-2) = 7^x- 7^x/7^2 = 7^x(1-1/49) = 7^x(48/49)

substituting LHS value
7^x(48/49)= 3 * 14^y
7^x =3* 49 * 14^y / 48 = 7^2 * 2^y * 7^y / 16

7^x = 7^2 * 2^y * 7^y / 2^4

thus both sides were integers then
y must be 4, cancelling out 2^y && 2^4
thus 7^x = 7^2 * 7^4 = 7^6
thus x=6

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Re: If x and y are integers and 7x−7x−2=3∗14y, what is the value of x?  [#permalink]

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New post 08 Mar 2017, 01:46
LHS: 7^x - 7^(x-2) = 7^(x-2){49 - 1} = 7^(x-2) x 48 = 7^(x-2) x 2^4 x 3
RHS: 3*14^y = 3 x 2^y x 7^y

equating the power of 2 we get y = 4
Equating the power of 7 we get x-2 = 4 or x = 6

Option D
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If x and y are integers and 7x−7x−2=3∗14y, what is the value of x?  [#permalink]

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New post 22 Jul 2017, 13:23
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This was a very challenging problem and really tests all your power laws. It's tempting to guess X =2 because if Y =1 the math works.

However we should solve to ensure. 14^Y is the same as (2*7)^Y which is the same as 2^Y X 7^Y. Anyways the solution follows. Also 7^(x+2) is the same as 7^X * 7^2. On top of all this they make it a 7^X * 7-2.
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Re: If x and y are integers and 7x−7x−2=3∗14y, what is the value of x?  [#permalink]

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New post 21 Jun 2019, 05:39
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Re: If x and y are integers and 7x−7x−2=3∗14y, what is the value of x?   [#permalink] 21 Jun 2019, 05:39
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