If d(x) denotes the no. positive divisors of x and d(d(d(x))), d(d(x)), d(x) and x can be arranged in 12 different ways. Find possible values of n(x), where n(x) is no. of prime factors of x. Also find possible values of x less than 100.

(In reply to computer exploration--lacking proof--spoilers by Charlie)

solution:

no. of ways of arrangement of these 4 nos. is 12 . This means that two of the nos. are equal. obviously only first two can be equal i.e d(d(d(x))) and d(d(x)).
As these two values are equal, so it can be concluded that they must be 2. thus d(x) must be a prime no. other than 2.
So, d(x) is an odd no. This means that 'x' is a perfect square, as only perfect squares can have odd numbers of factors.
and n(x) i.e no. of prime factors of 'x' will always be an even no.

Posted by rupesh on 2007-07-27 18:34:15