Ordered Pairs
Find all ordered pairs (A,B) such that:
A! = 1680 * (B!)
where "!" denotes the factorial function. For those unfamiliar, the factorial function is defined as the product of all integers less than or equal to the argument and greater than zero. For example, 6! = 6 * 5 * 4 * 3 * 2 * 1. In addition, the argument must be a non-negative integer (i.e. (6.5)! does not exist).
-- Brian Furtado
A! = 1680 * (B!)
where "!" denotes the factorial function. For those unfamiliar, the factorial function is defined as the product of all integers less than or equal to the argument and greater than zero. For example, 6! = 6 * 5 * 4 * 3 * 2 * 1. In addition, the argument must be a non-negative integer (i.e. (6.5)! does not exist).
-- Brian Furtado
Labels: SharedPuzzle
posted by ToM at
2:19 AM
5 Comments:
Too easy: 8! = 1680 * 4!
Ah, I've just re-read the question. You want ALL ordered pairs.
I think that the only other answer is 1680! = 1680 * 1679!
I agree that (8,4) and (1680,1679) are the only ordered pairs.
Method:
-Find all factors of 1680
-1(trivial), 2,3,5,7
-1680=2^4*3*5*7
7 and 5 are non-consecutive, so 6 must be inserted
-1680=5*(2*3)*7*(2^3)=5*6*7*8
revealing 8! and 4! as a pair
-no other consecutive series (2 or more numbers)using 2^4*3*5*7 can be constructed, eliminating other solutions
-1680! 1679! is another obvious pair
Cam
Yeah, this was a submission of mine... I guess it is a little easy but most of the people I gave it to missed the (1680,1679) ordered pair. I'll try to make it more difficult next time.
Hey chris and mo... I've added the answer to "The Three Laws" question
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