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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.


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.


Age 14 to 16 Challenge Level:

If factorial $100$ ($100!$) was rewritten as the product of its prime factors how many $2$s and how many $5$s would there be?
Along the way, confirm how many zeros are at the end of this large number and what the first digit to precede them is.

In the Teachers' Resources, Lyndon explains why he likes this problem.