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# M and N are two positive integers and each of them has 20 factors. If

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Joined: 25 Apr 2018
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Location: India
Concentration: Strategy, Finance
Schools: ISB '20, SMU '21
GMAT 1: 600 Q40 V33
M and N are two positive integers and each of them has 20 factors. If  [#permalink]

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21 Apr 2019, 07:37
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Difficulty:

65% (hard)

Question Stats:

44% (02:07) correct 56% (01:52) wrong based on 32 sessions

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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

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Re: M and N are two positive integers and each of them has 20 factors. If  [#permalink]

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21 Apr 2019, 11:00
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For the maximum value of r - s, let's find the maximum value of r and the minimum value of s.

Given that r is the total number of prime factors of M, and M is a positive integer that has 20 factors.
For the maximum value of r, let's assume that M = prime number $$p1^{1}p2^{1}p3^{4}$$. In such a case, the number of factors of M is 20 and the number of prime factors is 3. We cannot have more than 3 prime numbers for the number of factors to be 20.
Hence, the lowest possible value of r is 3.

Also, Given that s is the total number of prime factors of N, and N is a positive integer that has 20 factors.
For the minimum value of s, let's assume that N = prime number $$p^{19}$$. In such a case, the number of factors of N is 20 and the number of prime factors is 1.
Hence, the lowest possible value of s is 1.

Therefore, Max (r) - Min (s) = 3 - 1 = 2. So, the correct answer is option C. 2

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##### General Discussion
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M and N are two positive integers and each of them has 20 factors. If  [#permalink]

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21 Apr 2019, 11:01
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A number with 20 factors can be written as
20 = 2*2*5 or$$a^1 * b^1 * c^4$$
20 = 4*5 or$$a^3 * b^4$$
20 = 10*2 or $$a^9 * b^1$$
20 = 20 or $$a^19$$

where a, b, c are prime factors
Hence a number with 20 factors can have a maximum of 3 and a minimum of 1 prime factors
3-1 = 2
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Re: M and N are two positive integers and each of them has 20 factors. If  [#permalink]

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22 Apr 2019, 08:35
Hi axezcole92,

https://gmatclub.com/forum/rules-for-po ... 33935.html
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Re: M and N are two positive integers and each of them has 20 factors. If   [#permalink] 22 Apr 2019, 08:35
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