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N000ughty Thoughts

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

Mod 3

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

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.

Odd Stones

Age 14 to 16
Challenge Level

Hint for Check Point :

Turn $2$ - $8$ - $17$ into $4$ - $7$ - $16$

then $4$ - $7$ - $16$ into $3$ - $9$ - $15$

Hint for proving the odd one is impossible :

This example isn't the same thing but might give you a clue about the kind of thinking to try.

In a $4$ circle problem and using $26$ stones the distribution $1$ - $4$ - $7$ - $14$ cannot be turned into $3$ - $5$ - $7$ - $11$

To understand why notice that in the first there are two odd and two even numbers while in the second the numbers are all odd.

On a "move" one value goes up by $3$ and the others go down be one.

What will happen to odd and to even numbers?

The Odd Stones problem isn't about odd or even numbers but a similar kind of thinking could be useful.

Good luck!