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

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Novemberish

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.

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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.

Weekly Problem 32 - 2006

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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