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Counting Factors

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

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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.

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Oh! Hidden Inside?

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

What an Odd Fact(or)

Age 11 to 14 Challenge Level:

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 $5$?