In general, if you start with A and B, where A contains all of B's prime factors, then you can adjust B by scaling by A's factors. For example, if you start with A=30 and B = 20, you can keep scaling B by 2, 3, or 5 until the desired quotient is an integer. In this example, you have: (30 ^ (20*x)) / ((20*x) ^ 30) As you increase x, the numerator will grow faster than the denominator and eventually the quotient will be an integer. In this case, x=2 fails, but x=3 works, x=5 works, x=6 works, etc. Tom
Excellent!
------ Quoting Tom Karzes <karzes@sonic.net>:
A can have prime factors that B lacks. Example: A=6, B=32.
Tom
I've been playing with commutators lately, and the following puzzle turned up ...
Suppose A and B are integers > 1.
When is A^B / B^A an integer? What values are possible?
Clearly A and B must have the same prime divisors, but the primes can occur to different powers. A = 45, B = 75 is a suggestive example.
Rich
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun