There is a way to find d(n)
from the prime factorization of n; it has to do with the exponents, not
the primes, so it pays to use higher powers of smaller primes for
lots of divisors. See also problem
10. - Dan "the Man"
So 9 ^ (a - b) = (9^a)
/ (9^b) = 16 / (9^b) = 16 / (9^(3/2)) = 16 / (3^3) = 16/27.
9b) Can you find numbers
a =/= b such that a ^ b = b ^ a ?
Answer: Sure, 4^2 = 2^4 =
16, but did you know
there is a whole curve of (a,b) in the ab-plane with a^b = b^a
?! Use an implicit plot to graph x^y - y^x = 0 ; it goes through
any (a,a); also (2,4), (4,2), and a host of other points such
as (2.478,3) approx.
Check this : 2.478^3 :=:
3^2.478. (The :=: means
An answer of "no" might be
considered correct, if you couldn't find the numbers!
9c) If a $ b means a ^ b
- b ^ a , what is 4 $ 6 ?
Answer: 4^6 - 6^4 = 4096 - 1296 = 2800. Note a $ b is often positive if a < b. Is it
Is there a simple condition
for when a $ b > 0 ?
9d) Is 2 $ (3 $ 4) the same
as (2 $ 3) $ 4 ?
Answer: No. Here's
a proof that $ is "not associative" :