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

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Mod 3

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

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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.

396

Stage: 4 Challenge Level: Challenge Level:1

The four digits 5, 6, 7 and 8 are put at random in the spaces of the number below:

3 _ 1 _ 4 _ 0 _ 9 2

  • Estimate the probability that the answer will be a multiple of 396.
  • Calculate the probability and explain your answer.
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