What 3-digit number has greatest number of diff. factors?

[LilBee]

What three digit number has the greatest number of different factors? (Note: the number p^k where p is prime, has (k + 1) factors. Explain.)

I am stumped on this question. I was told that numbers with the same number at the front and at the end usually have the most. I tried this but worked out that 968 had more than 484 and 121 as they were factors of 968.

Is that the answer? Is there a formula to use? How do i explain the p^k?

Any help would be greatly appreciated!

 

[pka]

Applying the fundamental theorem of arithmetic we can factor the 3-digit numbers.
\(\displaystyle 900 = 2^2 \cdot 3^2 \cdot 5^2\) now any factor of 900 looks like \(\displaystyle 2^a \cdot 3^b \cdot 5^c \quad 0 \le a,b,c \le 2\) ; that is, each of a, b, & c can have any of three values: 0, 1, or 2.
Therefore, there are 27 factors of 900. Is that the largest?