Counting Factors

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

Repeaters

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.

Oh! Hidden Inside?

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

Indivisible

Stage: 3 Short Challenge Level:

Some students (fewer than $100$) are having trouble lining up  for a school production.

When they line up in $3$s, two people are left over.
When they line up in $4$s, three people are left over.
When they line up in $5$s, four people are left over.
When they line up in $6$s, five people are left over.

How many students are there in the group?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.
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