p, m, and n are positive integers greater than 1 such that p = m x n a

Question

p, m, and n are positive integers greater than 1 such that p = m x n and m and n do not have any common prime factors. If p has 25 factors, m has r factors, and n has s factors, what is the value of (r + s)?

A. 4
B. 8
C. 10
D. 16
E. 25

A. 4
B. 8
C. 10
D. 16
E. 25

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sujoykrdatta · Answer

p = m x n
Since p has 25 factors, p = a^24 (since 24+1 = 25) or a^4 × b^4 (since (4+1)(4+1) = 5×5 = 25)
Since m and n have no common prime factors and both are greater than 1, we cannot have p = a^24

Thus, p = a^4 × b^4 => m = a^4 and n = b^4 => m has 4+1 = 5 factors i.e. r = 5 and n also has 5 factors i.e. s = 5
=> r + s = 10

Answer C

Dereno · Answer

p,m,n are positive integers, which means the values are greater than 0.

But here p,m,n are greater than 1.

p = m * n

p has 25 factors.

m has r factors, and n has s factors.

Let’s take an example for easier understanding: 12 = 2^2 * 3 .

Where m =2 has (2+1)=3 factors = r.

n = 3 has (1+1)= 2 factors = s.

12 has 3*2 = 6 factors.

Let’s jump to the problem: 25 = r*s

The possible values of r*s = (25,1) (5,5).

The case (25,1) is not possible. We cannot have a term with 1 factor.

Hence only valid scenario is (5,5).

Thus, r+s = 5+5 =  10.

Option C.

razmanoraz · Answer

yo wassup, 
Shouldn't it be (x+1)(y+1)=25 and since 25 is 25x1 or 5x5 and p>1 there is only 5 x 5 so it's (x+1)=5 and (y+1)=5 so its x=4 and Y=4 so 4+4=8 ?

arushi118 · Answer

Property of Coprime Factors: Since m and n share no common prime factors, the number of factors of their product p is the product of their individual factors:

N(p) = N(m) * N(n)
 25 = r * s
Both m and n are greater than 1, so they must each have at least 2 factors (the number 1 and themselves). 
Therefore, r > 1 and s > 1.

The only factors of 25 greater than 1 are 5*5.
 r = 5 and s = 5

Calculate the sum: r + s = 5 + 5 = 10

Correct Answer: C

ExpertsGlobal5 · Answer

Video explanation:

https://www.youtube.com/watch?v=bTdDaJQM380

p, m, and n are positive integers greater than 1 such that p = m x n a

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p, m, and n are positive integers greater than 1 such that p = m x n a

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