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21 Feb 2023, 11:00
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What is the range of the set consisting of prime factors of \(m\), if \(m = 2^5*3^{11} - 9^{6} - 3^{11}\) ?
A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. \(5\)
A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. \(5\)
21 Feb 2023, 11:00
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Official Solution:
What is the range of the set consisting of prime factors of \(m\), if \(m = 2^5*3^{11} - 9^{6} - 3^{11}\) ?
A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. \(5\)
We can start by simplifying the expression for \(m\):
\(2^5* 3^{11} - 9^6 - 3^{11} = \)
\(=2^5*3^{11} - (3^2)^6 - 3^{11} = \)
\(=2^5 *3^{11} - 3^{12} - 3^{11}\)
Next, we can factor out \(3^{11}\) to obtain:
\(3^{11}(2^5 - 3 - 1) = \)
\(=3^{11} *28=\)
\(=3^{11} *2^2*7\)
Therefore, the set consisting of prime factors of \(m\) is {2, 3, 7}. The range of this set is (the largest term) - (the smallest term) = 7 - 2 = 5.
Answer: E
What is the range of the set consisting of prime factors of \(m\), if \(m = 2^5*3^{11} - 9^{6} - 3^{11}\) ?
A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. \(5\)
We can start by simplifying the expression for \(m\):
\(2^5* 3^{11} - 9^6 - 3^{11} = \)
\(=2^5*3^{11} - (3^2)^6 - 3^{11} = \)
\(=2^5 *3^{11} - 3^{12} - 3^{11}\)
Next, we can factor out \(3^{11}\) to obtain:
\(3^{11}(2^5 - 3 - 1) = \)
\(=3^{11} *28=\)
\(=3^{11} *2^2*7\)
Therefore, the set consisting of prime factors of \(m\) is {2, 3, 7}. The range of this set is (the largest term) - (the smallest term) = 7 - 2 = 5.
Answer: E
