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What is the units digit for the number 123^(456) ?


a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

A Biggy

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

Largest Expression

Age 14 to 16 Short
Challenge Level

Given that $x$ is positive and less than $1$, list the following five expressions from smallest to largest:

$x^2 + x$         $x^2$         $x^3$         $x^3 + x^2$          $x^4$
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.
You can find more short problems, arranged by curriculum topic, in our short problems collection.