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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

Answer: $x^4 \lt x^3 \lt x^2 \lt x^3 + x^2 \lt x^2 + x$

$x^2 + x \gt x^2$ because $x$ is positive

$x^3 = x^2\times x \lt x^2$ because $x\lt1$

$x^3 + x^2 \gt x^3, x^2$ because $x^3$ and $x^2$ are positive

$x^3 + x^2 = x\times\left(x^2 + x\right) \lt x^2 + x$ (or $x^3 + x^2 = x^2 + x^3 \lt x^2 + x$ because $x^3\lt x$)

And $x^4 =x\times x^3 \lt x^3$

So $x^4 \lt x^3 \lt x^2 \lt x^3 + x^2 \lt x^2 + x$

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.