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N000ughty Thoughts

How many noughts are at the end of these giant numbers?

DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

LCM Sudoku

Age 14 to 16 Challenge Level:

You can express $8, 9, 6, 4, 2, 5$ and $7$ as $2^3, 3^2, 2 \times 3, 2^2, 2, 5$ and $7$.

That means the LCM must contain $2^3, 3^2, 5$ and $7$ to ensure that it is a multiple of all the numbers.

So the LCM is $2^3 \times 3^2 \times 5 \times 7 = 2520$.