### 14 Divisors

What is the smallest number with exactly 14 divisors?

### 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.

##### Stage: 3 Challenge Level:

Powers of numbers behave in surprising ways.
Can you find convincing arguments that explain why all the statements below are true?

a) $2^{1}, 2^{2}, 2^{3},......, 2^{99}$ are never multiples of $10$.

b) $2^{1} + 3^{1}$, $2^{3} + 3^{3}$, $2^{5} + 3^{5}$, ......, $2^{99} + 3^{99}$ are all multiples of $5$.

c) $1^{99} + 2^{99} + 3^{99}$ is even

d) $1^{99} + 2^{99} + 3^{99} + 4^{99}$ is a multiple of $5$

e) $1^{99} + 2^{99} + 3^{99} + 4^{99} + 5^{99}$ is a multiple of $5$

f) $2^{99} + 3^{99} + 4^{99} + 5^{99} + 6^{99}$ is a multiple of $5$.

g) $3^{99} + 4^{99} + 5^{99} + 6^{99} + 7^{99}$ is a multiple of $5$.

h) $1^{x} + 2^{x} + 3^{x} + 4^{x} + 5^{x}$ is a multiple of $5$ when x is odd.

What other surprising results can you find? Can you explain why they are true?

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