Real Numbers for Math Olympiad: Complete Preparation Guide

Why This Matters

Real numbers form the deepest number theory topic in the school-to-Olympiad bridge. Euclid's division algorithm, the fundamental theorem of arithmetic, and irrationality proofs are all fair game.

For Class 10 students, Olympiad problems test your understanding of divisibility at a theoretical level. Can you use Euclid's algorithm creatively? Can you prove that a number is irrational?

Best Strategy

Common Pitfalls

Mistakes:

• Euclid's algorithm direction — Divide the larger by the smaller. Continue until remainder is 0.
• FTA uniqueness — The factorization is unique only up to the order of factors.
• Irrationality proof structure — Assume $\sqrt{p}$ is rational, derive that $p$ divides both numerator and denominator, contradicting coprimality.
• Decimal representations — Terminating decimals have denominators with only 2 and 5 as prime factors.

Practice Questions

Try these!

Question 1

Use Euclid's algorithm to find HCF of 867 and 255.

Solution: $867 = 255 \times 3 + 102$
$255 = 102 \times 2 + 51$
$102 = 51 \times 2 + 0$
HCF = 51.

Question 2

Prove that $\sqrt{5}$ is irrational.

Solution sketch: Assume $\sqrt{5} = \frac{a}{b}$ where $a, b$ are coprime. Then $5b^2 = a^2$, so 5 divides $a^2$, hence 5 divides $a$. Let $a = 5k$. Then $5b^2 = 25k^2$, so $b^2 = 5k^2$, meaning 5 divides $b$ too. Contradiction with coprimality.

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