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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.

Powerful Zeros

Age 14 to 16 Short
Challenge Level

Answer: 6 zeros

Multiplying by $10$, or multiples/powers of $10$, makes zeros, so find powers of $10$
    $3^4\times 4^5\times 5^6$
$=3^4\times 2^{10}\times 5^6$
$=3^4\times 2^4\times 2^6 \times 5^6$
$=\underbrace{6^4}_{\text{no zeros}}\times\underbrace{10^6}_{\text{six zeros}}$

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