Stage: 5 Challenge Level: 
Use the fact that:
| $x^2-y^2$ | = | $(x-y)(x+y)$ |
| $x^3+y^3$ | = | $(x+y)(x^2-xy+y^2)$ |
to find the highest power of $2$ and the highest power of $3$ which divide $5^{36}-1$.
Factors and multiples. Long problems. Modulus arithmetic. Quadratic equations. Pythagoras' theorem. Factorisation (algebraic). Working systematically. Mathematical reasoning & proof. Identities. Index notation/Indices.
Published June 1998.
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