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.
How many integers do you need to ensure that the product of all the differences is divisible by $5$?
Some students may go on to investigate this context more thoroughly, including posing and pursuing their own questions. For example: What about divisibility by $4$ and $6$, and then more generally?
Odd Stones and Take Three from Five might provide suitable follow-up problems.