Is there an efficient way to work out how many factors a large number has?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Find the number which has 8 divisors, such that the product of the divisors is 331776.

Find the highest power of 11 that will divide 1000! exactly.

[Factorial 1000, written 1000!, is the product of the first 1000 whole numbers 1 x 2 x 3 x ... x 1000]