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25 Oct 2021, 06:50
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
59% (02:00) correct
41%
(02:23)
wrong
based on 219
sessions
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If P is a positive integer and the least common multiple of \(21^{14}, 9^8 \), and P is \( 21^{16}\) , then how many different values can P take?
A. 8
B. 14
C. 15
D. 16
E. 17
A. 8
B. 14
C. 15
D. 16
E. 17
25 Oct 2021, 07:04
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ktzsikka
Breaking Down the Info:
The LCM is \(21^{16} = 3^{16}*7^{16}\). In order to build this LCM, we need the inputs \(3^{16}\) and \(7^{16}\).
We can see that we already have \(9^8 = 3^{16}\), and we are missing \(7^{16}\). Thus P must incorporate the factor \(7^{16}\).
Next, we are allowed to add factors of 3 freely into P as that does not contribute to the LCM (only the highest powers contribute). Hence, the info that \(21^{14}\) is an ingredient for the LCM is not useful.
The power of 3 for P can be from 0 to 16, so that is 17 numbers.
Answer: E
25 Oct 2021, 07:46
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ktzsikka
\(21^{14}=3^{14}*7^{14}\)
\(9^8=3^{16}\)
LCM of \( 3^{14}*7^{14}\) and \( 3^{16}\) is \( 3^{16}*7^{14}\)
Now,
\(21^{16}=3^{16}*7^{16}\)
LCM of \( 3^{16}*7^{14}\) and \(P\) is \( 3^{16}*7^{16}\)
Minimum value of P: \(7^{16}\)
Maximum value of P: \( 3^{16}*7^{16}\)
Thus P can be \( 3^{x}*7^{16}\), where x is 0 to 16 => Total 17 numbers.
E
