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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.

Different by One

Age 14 to 16
Challenge Level

In the original example, the lines were made from rods that differed in length by 2:
light green rods (3) and yellow rods (5).

You could start by creating lines made from
pink (4) and dark green (6)
yellow (5) and black (7)
dark green (6) and brown (8)
black (7) and blue (9)...

Which pairs can be used to create lines that differ by 1?
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