Prime structure
Objective. Apply fundamental factorisation and prime tests.
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- 2. Worked example
- 3. Practice
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Learn: the key idea
Prime structure concerns how integers factorise uniquely into primes, the building blocks of multiplication, as guaranteed by the fundamental theorem of arithmetic. Every integer greater than 1 has a unique factorisation up to ordering, and primality can be tested by trial division up to √n or by more efficient probabilistic tests like Miller–Rabin for large numbers. Knowing the prime factorisation lets one compute gcds, lcms, and the number of divisors directly from exponents. For example, 360 = 2³×3²×5, so it has (3+1)(2+1)(1+1) = 24 divisors in total.
Worked example
Find the prime factorisation structure of 198 and state the number of its positive divisors.
- 1Divide by 2: 198 = 2×99
- 2Factorise 99: 99 = 3²×11
- 3So 198 = 2×3²×11
- 4Divisors = (1+1)(2+1)(1+1) = 12
Answer: 12
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Start practisingCommon mistake. Students test primality of n by checking divisibility only up to n/2; it suffices and is far faster to check only up to √n, since any factor pair must include one factor at most √n.
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