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For What?

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

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Ordered Sums

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.

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Triangles Within Triangles

Can you find a rule which connects consecutive triangular numbers?

In Particular

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Write 100 as the sum of two positive integers, one divisible by 7 and the other divisible by 11. Use your answer to find formulas giving all the solutions of the following equation where x and y are integers.

$$7x + 11y =100$$