Thirty Six Exactly

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

Even So

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

Power Crazy

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

Powerful Zeros

Stage: 3 Short Challenge Level:

6

We need to write $3^4\times 4^5\times 5^6$ in the form $a\times 10^n$ where $a$ is not a multiple of 10.

$3^4\times 4^5\times 5^6=3^4\times 2^{10}\times 5^6=3^4\times 2^4\times 2^6 \times 5^6=3^4\times 2^4\times 10^6.$

Hence the number ends in six zeros.

This problem is taken from the UKMT Mathematical Challenges.