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A rational number problem (Posted on 2006-10-02) Difficulty: 3 of 5
Determine the total number of rational numbers of the form m/n, where m and n are positive integers such that:

(A) m/n lies in the interval (0, 1); and

(B) m and n are relatively prime; and

(C) mn = 25!

NOTE: "!" denotes the factorial symbol, where n! = 1*2*3*......*(n-1)*n

See The Solution Submitted by K Sengupta    
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Solution Got it | Comment 1 of 12

I believe there are 256 such numbers.

There are 9 prime factors of 25! = 2^22*3^10*5^6*7^3*11^2*13*17*19*23

Each of these prime factors can only be a factor of m or n, so there are 2^9 ways to split m and n so they are relatively prime.  In only half of these is m<n so the solution is 2^9/2=2^8=256

If we ignore condition (B) there are 2^(23*11*7*4*3*2*2*2*2)/2=2^340031 solutions.

If we ignore both (A) and (B) there are 2^340032 solutions


  Posted by Jer on 2006-10-02 14:35:17
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