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.

Missing Digit

Stage: 3 Short Challenge Level:
A test for divisibility by 11 is to add alternate digits:

1 + 3 + * + 7 = 11 + *; 2 + 4 + 6 + 8 = 20.

If the original number is a multiple of 11 then these two totals will be the same or will differ by a multiple of 11. In this case, 11 + * = 20 gives * = 9.

Or, you can solve it without knowing a rule as follows:

1234*678 = 12340678 + 1000* = (11 x 1121879 +9) + 11 x 90* + 10*

and hence is divisible by 11 if and only if 10* + 9 is divisible by 11. So * = 9.

This problem is taken from the UKMT Mathematical Challenges.
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