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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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re: Got it | Comment 2 of 12 |
(In reply to Got it by Jer)

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

Why two to the power? There are 23 choices of how many 2's factor into the first factor, 11 choices of how many 3's, etc. for just 340031.  Then we have to halve that, which is 170016. And that's the answer ignoring condition B.

Then if you ignore A, just double this, as the second need no longer be larger than the first, to 340031 again. (Actually, you gave the same 2^340032 for both situations.)

  Posted by Charlie on 2006-10-02 15:04:58
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