Burt Kaliski Jr. (@modulomathy)
4/5/12 8:25 AM
#math trivia for #April5: #96 ties record for prime factors counting repeats (6), but doesn’t match record for divisors (12). Why not?
The six prime factors of 96, counting repeats, are 2, 2, 2, 2, 2, and 3. This ties 64, which is the product of six 2s.
The record for divisors so far is held by 60, which has 12. However, as it turns out, 96 ties this record as well. The problem statement was incorrect.
The point I was trying to get at is that more prime factors doesn’t necessarily mean more divisors. Suppose a number N is a product of prime powers
N = p1^a1 * p2^a2 * … * pk^ak
The number of prime factors, counting repeats, is the sum
a1 + a2 + … + ak
The number of divisors is the product
(a1+1)*(a2+1)* … * (ak+1)
When N = 96, there are two primes, and a1 = 5, a2 = 1. The first value is thus 6 and the second is 12. When N = 60, there are three primes, and a1 = 2, a2 = 1, a3 = 1. The second value is again 12. To get a larger number of divisors without the number itself being too large, either the initial a’s (corresponding to smaller primes) should be increased, or there should be another prime. An example of the first approach is N = 120, which has 16 divisors. An example of the second is N = 210, which also has 16. Both have five prime factors including repeats — less than 96 does even though both have more divisors.