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Multiplication Magic

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). The question asks you to explain the trick.

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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}.

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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.

Number Rules - OK

Age 14 to 16 Challenge Level:

image of colourful numbers.Can you convince me of each of the following?

  • The pattern below continues forever: $$8^2 = 7^2 + 7 + 8$$ $$9^2 = 8^2 + 8 + 9$$
  • If a square number is multiplied by a square number the product is ALWAYS a square number.
  • No number terminating in $2, 3, 7$ or $8$ is a perfect square.