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
(In reply to re: Got it
by Dej Mar)
Jer's 2^9/2 already includes the possibility of only 1 as one of the two numbers m or n.
One of the 2's that make up the 2^9 is the decision to include or exclude the 2^22 that goes into the factorization of 25!. Another of the 2's represents the decision to include or exclude the 3^10 that goes into the factorization of 25!. Another represents including or excluding the 5^6, etc.
Note that one of the 2^9 possibilities is to exclude them all from m; that leaves just a factor of 1 to be n. The possibility to include them all in m leaves the other number (n) as 1. The division by 2 reduces that to just the 1/25! and excudes the 25!/1.
Posted by Charlie
on 2006-10-04 00:08:54