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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.

There's Always One Isn't There

Age 14 to 16
Challenge Level

Try plenty of examples :

Maybe start with other numbers and take multiples but continue dividing by 9.

Next use other numbers again, taking multiples of each as before but now divide by 7.

Then again use different numbers, take multiples and divide by 8 - be sure to include 12 as on of the numbers you try multiples of.

What happens?