Number theory · Grade 12

In Grade 12 you learn: modular arithmetic.

Number theory studies the whole numbers: divisibility, primes, remainders and the patterns they make. It was once the purest of pure mathematics. Now it secures online payments, because modular arithmetic makes some calculations easy one way and practically impossible to reverse.

Key facts

• a mod m is the remainder when a is divided by m. a ≡ b (mod m) means m divides a − b.
• Euclidean algorithm: gcd(a, b) = gcd(b, a mod b). Repeat until the remainder is 0.
• Bézout: there are integers x and y with ax + by = gcd(a, b). So a has an inverse mod m exactly when gcd(a, m) = 1.
• If n = pᵃ × qᵇ (p, q prime), n has (a + 1)(b + 1) positive divisors.
• Fermat’s little theorem: if p is prime and does not divide a, then a^(p−1) ≡ 1 (mod p).
• lcm(a, b) = ab ÷ gcd(a, b).

Worked examples

Other years

• Number theory · Grade 11