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If a and b are both positive integers greater than 1 and a^b = a^(11b)

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If a and b are both positive integers greater than 1 and a^b = a^(11b)  [#permalink]

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New post Updated on: 17 Dec 2018, 02:54
1
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A
B
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D
E

Difficulty:

  65% (hard)

Question Stats:

57% (01:25) correct 43% (01:44) wrong based on 47 sessions

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If \(a\) and \(b\) are both positive integers greater than 1 and \(a^b\) = \(a^{(11b - 60)}\), what is the value of \(a*b\) ?


(1) \(a^ 2 = 7|a|\)

(2) \(|a| = 7\)

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Originally posted by Helium on 17 Dec 2018, 02:13.
Last edited by Helium on 17 Dec 2018, 02:54, edited 4 times in total.
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If a and b are both positive integers greater than 1 and a^b = a^(11b)  [#permalink]

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New post Updated on: 17 Dec 2018, 04:00
b=6 from the equation a^b=a^(11b-60)

1. a^2=7|a|
a*a=7*|a|
implies, a can be +7 or -7, since it is given that a and be are positive, making a*b +ve definite .
sufficient.
2. similarly, a can be +7 or -7, a= +7 . a*b is +ve definite value.


Hence 'D' is correct.

Originally posted by jashandeep2332 on 17 Dec 2018, 03:47.
Last edited by jashandeep2332 on 17 Dec 2018, 04:00, edited 1 time in total.
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If a and b are both positive integers greater than 1 and a^b = a^(11b)  [#permalink]

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New post 17 Dec 2018, 03:49
Harshgmat wrote:
If \(a\) and \(b\) are both positive integers greater than 1 and \(a^b\) = \(a^{(11b - 60)}\), what is the value of \(a*b\) ?


(1) \(a^ 2 = 7|a|\)

(2) \(|a| = 7\)


Here is my take on this question.

Given \(a^b\) = \(a^{(11b - 60)}\)
b=11 b - 60
b=6

Now the question becomes a*6 = ?

1) \(a^ 2 = 7|a|\)

\(a^ 4 - 49 a^2 = 0\)
\(a^ 2(a^2 - 49) = 0\)

Solving which will give value of a as a= 0, 7 and -7

But since it is given that a is a positive number we can ignore -42 and 0, giving us the answer as 42.

Which is sufficient by its own.

2) \(|a| = 7\)
will give value of a as 7 and -7

Here we will get multiple values of a*6 = 42 or -42

But since it is given that a is a positive number we can ignore -42, giving us the answer as 42.

Which is sufficient by its own.

Giving the final answer as
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If a and b are both positive integers greater than 1 and a^b = a^(11b)  [#permalink]

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New post 17 Dec 2018, 12:44
Doubted myself very much with the absolute value, but the prompt says, a > 0, and b > 0, so for the purposes of the question, I only focused on positive possibilities.

from the given equation: a^b = a ^(11b -60), the same base, so can focus on B = 11B - 60 -> -10B = -60 -> 10b = 60 - > b = 6

We get B = 6

1) a^2 = 7*|a|, again only focused on (+), so for statement 1 to be true, a can only be 7. 49 = 49.

plug back into the question 7*6 = 42

2) |a| = 7, no need to solve since again only concerned with the (+), so sufficient.

My answer is D
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Re: If a and b are both positive integers greater than 1 and a^b = a^(11b)  [#permalink]

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New post 17 Dec 2018, 13:23
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Guys, I see you excellent and detailed explanations, and you are correct.

there is one hint that I wish to add which could save you time.

Memorize that \(\frac{a^2}{|a|} = |a|\)

so in statement 1- \(a^2 = 7|a|\), we can simply divide both sides by |a| and simplify it to \(|a| = 7\)

I wish this is helping, not confusing.
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Re: If a and b are both positive integers greater than 1 and a^b = a^(11b) &nbs [#permalink] 17 Dec 2018, 13:23
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