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09 Jun 2020, 05:07
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Difficulty:
95%
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Question Stats:
47% (02:38) correct
53%
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based on 106
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The product of the largest two factors of a positive integer n is 16875. What is the difference between the largest positive integer that divides √n and the smallest odd factor greater than 1 of √n?
A. 2
B. 12
C. 13
D. 222
E. 223
Are You Up For the Challenge: 700 Level Questions
A. 2
B. 12
C. 13
D. 222
E. 223
Are You Up For the Challenge: 700 Level Questions
09 Jun 2020, 06:26
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\(16875= 5^4*3^3\)
Since the prime factors of 16875, are only 3 and 5, prime factors of n are also 3 and 5.
Largest factor of n is n and second largest is n/3.
so \(\frac{n^2}{3} =16875 = 5^4*3^3\)
\(n^2 = 5^4*3^4\)
\(√n = 5*3 =15\)
the largest positive integer that divides √n = 15
the smallest odd factor greater than 1 of √n = 3
difference = 15-3=12
Since the prime factors of 16875, are only 3 and 5, prime factors of n are also 3 and 5.
Largest factor of n is n and second largest is n/3.
so \(\frac{n^2}{3} =16875 = 5^4*3^3\)
\(n^2 = 5^4*3^4\)
\(√n = 5*3 =15\)
the largest positive integer that divides √n = 15
the smallest odd factor greater than 1 of √n = 3
difference = 15-3=12
Bunuel
General Discussion
10 Jun 2020, 04:28
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\(16875 = 3^3 * 5^4\)
Largest factor of integer N will be N itself. Let the second largest be X.
So, \(N*X = 16875 = 3^3 * 5^4\) ------(1)
N = N*1 = X*P, where P is the second lowest factor of N.
=> \(N^2 = N*X*P\)
=> we need to multiply a 3 to obtain square of N.
=> \(N = 3^2*5^2\)
=> \(N = 15\)
Largest factor of 15 = 15
Second largest odd factor of 15 = 3
Difference = 12.
Largest factor of integer N will be N itself. Let the second largest be X.
So, \(N*X = 16875 = 3^3 * 5^4\) ------(1)
N = N*1 = X*P, where P is the second lowest factor of N.
=> \(N^2 = N*X*P\)
=> we need to multiply a 3 to obtain square of N.
=> \(N = 3^2*5^2\)
=> \(N = 15\)
Largest factor of 15 = 15
Second largest odd factor of 15 = 3
Difference = 12.
