The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
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?
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?
Every Maundy Thursday the reigning monarch distributes 'Maundy money' to equal numbers of men and women. One year at Newcastle 64 men and 64 women each received 64p, one penny for each year of the Queen's life. Written as a power of 2, what was the total amount distributed in pence?
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.