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.

What can you say about the patterns in the last digits of powers of $2$, $3$, $4$ etc?

How can you use these patterns to say what the last digits are of the numbers raised to the power $99$?

Now can you say whether the sum is divisible by $5$?