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If a and b are both positive integers greater than 1 and a^b = a^(11b) 17 Jan 2020, 02:18
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D
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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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D
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If a and b are both positive integers greater than 1 and a^b = a^(11b) 17 Jan 2020, 06:38
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If the base a is greater than 1, then the equation in the stem lets us find b (we can set the exponents equal and find b = 6). Since a is positive, the absolute values in each statement do nothing, so we can ignore them, and solve for a using each statement (in Statement 1, we ignore the solution a = 0 since we know a is positive); in each case we find a = 7. With the values of both a and b we can answer the question, so the answer is D.
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If a and b are both positive integers greater than 1 and a^b = a^(11b) 21 Jan 2020, 09:37
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Bunuel
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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\(a^b=a^{(11b - 60)}…a^b-a^{(11b - 60)}=0\)
\(a^b(1-a^{(10b - 60)}=0…a^b=0…or…1-a^{(10b - 60)}=0\)
\((a,b)>1:a^b≠0…1-a^{(10b - 60)}=0…10b-60=0…b=6\)

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

\(a^ 2 = 7|a|…(a>1)…a^2-7a=0…a(a-7)=0\)
\(a=0…or…a-7=0…(a>1:a≠0)…a=7…ab=7*6\)

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

\(a>1:|a|=7…a=7…ab=7*6\)

Ans (D)
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If a and b are both positive integers greater than 1 and a^b = a^(11b) 25 Jan 2020, 11:23
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Bunuel
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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Analyzing the question:
Since we are given \(a^b=a^{(11b - 60)}\) while knowing \(a\) is greater than 1, the only way for this to be true is to let the exponents equate since with a base of \(a > 1\), \(a^x\) is an increasing function. \(b = 11b - 60\) and \(b = 60/10 = 6\), we would like to know the value of \(a\) now.

Statement 1:
\(a^ 2 = |a|^2 = 7|a|\)
\(|a| = 7\)
a = 7 or a = -7, however \(a\) must be positive so a = 7 is the only case we can take. Sufficient.

Statement 2:
Same as statement 1, sufficient.

Ans: D
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If a and b are both positive integers greater than 1 and a^b = a^(11b) 25 Jan 2020, 12:46
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Bunuel
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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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Since a is positive and \(a^b = a^{11b-60}\), we have \(b = 11b - 60\) or \(b = 6\).

Since we have 2 variables (\(a\) and \(b\)) and 1 equation(\(b=6\)), D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
\(a^2 = 7|a|\)
⇔ \(a^2 = 7a\), since \(a>0\)
⇔ \(a^2 - 7a = 0\)
⇔ \(a(a-7) = 0\)
⇔ \(a = 0\) or \(a = 7\)
Since \(a\) is positive, we have \(a = 7\)

Thus, we have \(a \cdot b = 7 \cdot 6 = 42\).

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

\(|a| = 7\)
⇔ \(a = ±7\)
Since a is positive, \(a = 7\).

Thus, we have \(a \cdot b = 7 \cdot 6 = 42\).

Since condition 2) yields a unique solution, it is sufficient.

Therefore, D is the answer.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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If a and b are both positive integers greater than 1 and a^b = a^(11b) 10 Aug 2022, 10:03
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