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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}.


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.

Latin Numbers

Can you create a Latin Square from multiples of a six digit number?

Number Rules - OK

Age 14 to 16
Challenge Level

For each of the cases below, try some numerical examples to convince yourself that each statement is true.

Then try to provide convincing pictorial and/or algebraic arguments that they are always true.

  1. Two consecutive numbers add to give an odd number
  2. The product of two consecutive numbers is even
  3. The sum of four consecutive numbers is never a multiple of $4$
  4. Two odd numbers add to give an even number
  5. The pattern below continues forever: $$7^2=6^2 + 6 + 7 $$ $$8^2 = 7^2 + 7 + 8$$ $$9^2 = 8^2 + 8 + 9$$
  6. Squaring an odd number always gives an odd number
  7. If a square number is multiplied by a square number the product is a square number

Final Challenge

Can you discover any other number rules and provide convincing arguments that they are always true?

Did you know ... ?

Although number theory - the study of the natural numbers - does not typically feature in school curricula it plays a leading role in university at first year and beyond. Having a good grasp of the fundamentals of number theory is useful across all disciplines of mathematics. Moreover, problems in number theory are a great leisure past time as many require only minimal knowledge of mathematical 'content'.

We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.