\(12^7 = a^b*c\)
=\((2^2*3)^7\)
=\(2^{14}*3^7\)
Statement (1) a is prime. a can be either 2 or 3
b can be 2 for c= \(2^{12} *3^7\),
b can be 3 for c=\( 2^{11} *3^7\),
.....so on
Not sufficient
Statement (2) c is not divisible by 2.C is odd and greater than 1
check possible values of c
C=3 \(a^b=2^{14}*3^6\) Not possible for integral values of a and b
same for other powers of 3 less than 7
C will be equal to \(3^7\); \(a^b=2^{14}\) or \(a^b=4^{7}\)
i.e b can be 14 or 7 etc
Not sufficient
Combined
C is odd and a is prime so b = 14
(C) is the correct answer IMO
Bunuel wrote:
If a, b, and c are integers greater than 1, and \(12^7 = a^b*c\), what is the value of b ?
(1) a is prime.
(2) c is not divisible by 2.
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