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### Number and algebra

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# Number Rules - OK

### Final Challenge

*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'.
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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.**

- Two consecutive numbers add to give an odd number

- The product of two consecutive numbers is even

- The sum of four consecutive numbers is never a multiple of $4$

- Two odd numbers add to give an even number

- The pattern below continues forever: $$7^2=6^2 + 6 + 7 $$ $$8^2 = 7^2 + 7 + 8$$ $$9^2 = 8^2 + 8 + 9$$
- Squaring an odd number always gives an odd number

- If a square number is multiplied by a square number the product is a square number

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

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.*

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.