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More Mods

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

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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.

Multiplication Magic

Age 14 to 16 Challenge Level:

111, 222, 333,... etc are all divisible by 37.

A second hint is to use 1000= 999+1 when breaking the number into blocks of 3 digits