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27 Oct 2023, 23:04
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Difficulty:
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68% (01:34) correct
32%
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The integers 6, 9, 10, and 25 are each factors of the positive integer k. Which of the following integers must also be a factor of k?
I. 54
II. 60
III. 90
A. III only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III
I. 54
II. 60
III. 90
A. III only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III
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27 Oct 2023, 23:24
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edwin.que
LCM of (6, 9, 10 , 15) = \(3^2 * 2 * 5^2\)
Hence, the least possible value of k = \(3^2 * 2 * 5^2\)
i. 54 → \(3^3 * 2\)
We know that \(3^2\) is a factor of \(k\), however, we cannot guarantee that \(3^3\) is a factor of k. Hence, Eliminate this option.
ii.60 → \(3 * 5 * 2^2\)
We know that \(2\) is a factor of \(k\), however, we cannot guarantee that \(2^2\) is a factor of k. Hence, Eliminate this option.
iii.90 → \(3^2 * 5 * 2\)
\(3^2 * 5 * 2\) can be a factor of \(k\). Let's keep this option.
III only.
Option A
General Discussion
27 Oct 2023, 23:28
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To determine which of the given integers must also be factors of k, we need to find the least common multiple (LCM) of the factors 6, 9, 10, and 25. Any integer that is a factor of this LCM will also be a factor of k.
Prime factorization of each integer:
6 = 2 * 3
9 = 3^2
10 = 2 * 5
25 = 5^2
Now, let's find the LCM by taking the highest power of each prime factor:
LCM = 2^1 * 3^2 * 5^2
Calculate the LCM:
LCM = 2 * 9 * 25 = 450
So, the LCM of 6, 9, 10, and 25 is 450.
Now, we need to check which of the given integers, 54, 60, and 90, are factors of 450.
i. 54 = 3^3 * 2
We know that 2^2 is a factor of k, but we cannot guarantee that 3^3 is a factor of k. So, we can eliminate this option.
ii. 60 = 3 * 5 * 2^2
We know that 2^2 is a factor of k, but we cannot guarantee that 3^2 is a factor of k. So, we can eliminate this option.
iii. 90 = 3^2 * 5 * 2
The prime factorization of 90 is 3^2 * 5 * 2, which includes 2^1, 3^2, and 5^1. Since 2^1, 3^2, and 5^1 are all factors of k, we can be sure that 90 is also a factor of k.
So, following your analysis, the only integer that must be a factor of k is iii. 90.
The correct answer is:
A. III only
Prime factorization of each integer:
6 = 2 * 3
9 = 3^2
10 = 2 * 5
25 = 5^2
Now, let's find the LCM by taking the highest power of each prime factor:
LCM = 2^1 * 3^2 * 5^2
Calculate the LCM:
LCM = 2 * 9 * 25 = 450
So, the LCM of 6, 9, 10, and 25 is 450.
Now, we need to check which of the given integers, 54, 60, and 90, are factors of 450.
i. 54 = 3^3 * 2
We know that 2^2 is a factor of k, but we cannot guarantee that 3^3 is a factor of k. So, we can eliminate this option.
ii. 60 = 3 * 5 * 2^2
We know that 2^2 is a factor of k, but we cannot guarantee that 3^2 is a factor of k. So, we can eliminate this option.
iii. 90 = 3^2 * 5 * 2
The prime factorization of 90 is 3^2 * 5 * 2, which includes 2^1, 3^2, and 5^1. Since 2^1, 3^2, and 5^1 are all factors of k, we can be sure that 90 is also a factor of k.
So, following your analysis, the only integer that must be a factor of k is iii. 90.
The correct answer is:
A. III only
