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M and N are two positive integers and each of them has 20 factors. If r is the total number of prime factors of M and s is the total number of prime factors of N, then what is the maximum value of r-s?
(A) 0
(B) 1
(C) 2
(D) 6
(E) 26
(A) 0
(B) 1
(C) 2
(D) 6
(E) 26
28 Nov 2022, 10:16
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20 = 2 * 5 * 2
So we can have three prime factors contributing to a total factor of 20, or a single prime factor contributing to a total factor of 20.
M = \(p_1 * p_2^4*p_3\) | r = 3
N = \(p^{19}\) | s = 1
\(p, p_1, p_2,\) and \(p_3 \) are prime numbers
Hence maximum r - s= 3 - 1 = 2
Option C
So we can have three prime factors contributing to a total factor of 20, or a single prime factor contributing to a total factor of 20.
M = \(p_1 * p_2^4*p_3\) | r = 3
N = \(p^{19}\) | s = 1
\(p, p_1, p_2,\) and \(p_3 \) are prime numbers
Hence maximum r - s= 3 - 1 = 2
Option C
06 Feb 2024, 22:44
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Given: M and N are two positive integers and each of them has 20 factors.
Asked: If r is the total number of prime factors of M and s is the total number of prime factors of N, then what is the maximum value of r-s?
20 = 1*20 = 2*10 = 4*5 = 2*2*5
Maximum value of r = 3 when M is of the form = p1ˆ * p2ˆ * p3ˆ4
Minimum value of s = 1 when N is of the form = pˆ19
Maximum value of r - s = 3 - 1 = 2
IMO C
Asked: If r is the total number of prime factors of M and s is the total number of prime factors of N, then what is the maximum value of r-s?
20 = 1*20 = 2*10 = 4*5 = 2*2*5
Maximum value of r = 3 when M is of the form = p1ˆ * p2ˆ * p3ˆ4
Minimum value of s = 1 when N is of the form = pˆ19
Maximum value of r - s = 3 - 1 = 2
IMO C
