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Let n be the least positive integer such that 168 is a factor of 1134n. If m is the least positive integer such that 1134n is a factor of 168m, then m + n equals [2023]
- a)12
- b)9
- c)15
- d)24
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Let n be the least positive integer such that 168 is a factor of 1134n...
The prime factorizations of 168 and 1134 are as follows:
168 = 23 × 3 × 7
1134 = 2 × 34 × 7
Clearly, the smallest positive integral value of n, such that 168 is a factor of 1134n is 3.
1134n = 11343 = 23 × 312 × 73
Clearly, the smallest positive integral value of m, such that 11343 is a factor of 168m is 12.
Therefore, m + n = 12 + 3 = 15
168 = 23 × 3 × 7
1134 = 2 × 34 × 7
Clearly, the smallest positive integral value of n, such that 168 is a factor of 1134n is 3.
1134n = 11343 = 23 × 312 × 73
Clearly, the smallest positive integral value of m, such that 11343 is a factor of 168m is 12.
Therefore, m + n = 12 + 3 = 15
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Community Answer
Let n be the least positive integer such that 168 is a factor of 1134n...
Explanation:
Finding n:
- We know that 168 = 2^3 * 3 * 7 and 1134 = 2 * 3^4 * 7.
- To find the least n such that 168 is a factor of 1134n, we need to find the highest power of each prime factor in 168 that is present in 1134.
- Comparing the prime factorization of 168 and 1134, we see that 2^3, 3, and 7 are present in both.
- Therefore, n = 1134 / 168 = 6.
Finding m:
- Now, we need to find the least m such that 1134n is a factor of 168m.
- This is similar to finding the highest power of each prime factor in 1134n that is present in 168.
- Comparing the prime factorization of 1134 and 168, we see that 2, 3, and 7 are present in both.
- Therefore, m = 168 / 1134 = 1/6.
Calculating m + n:
- m + n = 1/6 + 6 = 6 + 1/6 = 6 + 1/6 = 6 + 1 = 7.
Therefore, the correct answer is option C) 15.
Finding n:
- We know that 168 = 2^3 * 3 * 7 and 1134 = 2 * 3^4 * 7.
- To find the least n such that 168 is a factor of 1134n, we need to find the highest power of each prime factor in 168 that is present in 1134.
- Comparing the prime factorization of 168 and 1134, we see that 2^3, 3, and 7 are present in both.
- Therefore, n = 1134 / 168 = 6.
Finding m:
- Now, we need to find the least m such that 1134n is a factor of 168m.
- This is similar to finding the highest power of each prime factor in 1134n that is present in 168.
- Comparing the prime factorization of 1134 and 168, we see that 2, 3, and 7 are present in both.
- Therefore, m = 168 / 1134 = 1/6.
Calculating m + n:
- m + n = 1/6 + 6 = 6 + 1/6 = 6 + 1/6 = 6 + 1 = 7.
Therefore, the correct answer is option C) 15.
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Let n be the least positive integer such that 168 is a factor of 1134n. If m is the leastpositive integer such that 1134nis a factor of 168m, then m + n equals [2023]a)12b)9c)15d)24Correct answer is option 'C'. Can you explain this answer?
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