Sec 1 Mathematics Primes, HCF & LCM notes

A factor of a number divides it exactly, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. Counting factors sorts whole numbers (greater than 1) into two families:

• 2 factors

  Prime numbers

  Exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13.

• 3 or more factors

  Composite numbers

  More than two factors. Examples: 4, 6, 8, 9, 10, 12.

• 1 factor

  The number 1

  Only one factor — itself — so it is neither prime nor composite.

The primes up to 50 are worth memorising: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Two quick tests catch most composites: a number is divisible by 2 if its last digit is even, and divisible by 3 if its digit sum divides by 3 (51 → 5 + 1 = 6, so 51 = 3 × 17 is composite). A number ending in 0 or 5 divides by 5.

A prime factorisation writes a number as a product of primes only. Two common layouts work equally well: a factor tree (split into any two factors, then keep splitting the composites) or a ladder (divide repeatedly by the smallest prime that works).

1. 1Split

   Split 72 into any two factors, e.g. 72 = 8 × 9.

2. 2Keep splitting composites

   8 = 2 × 4, 4 = 2 × 2 and 9 = 3 × 3. Stop when every branch ends in a prime.

3. 3Collect and count

   72 = 2 × 2 × 2 × 3 × 3. Count each prime.

4. 4Write in index notation

   72 = 2³ × 3² — the small number says how many times each prime multiplies.

Index notation is just a shorthand: 2³ means 2 × 2 × 2, and 5² means 5 × 5. Whichever pair of factors you split off first, you land on the same prime product — for 72, starting with 72 = 2 × 36 gives 2³ × 3² as well.

A perfect square is a whole number times itself, like 36 or 169. A perfect cube is a whole number used three times in a product, like 27 (= 3 × 3 × 3) or 1000 (= 10 × 10 × 10). The square root of a number undoes squaring: since 13² = 169, the square root of 169 is 13. At Sec 1, the square root symbol means the positive root.

Prime factorisation finds roots without a calculator. In a perfect square every prime appears an even number of times: 324 = 2² × 3⁴ = (2 × 3 × 3) × (2 × 3 × 3), so its square root is 2 × 3 × 3 = 18. In a perfect cube every prime appears a multiple of three times: 216 = 2³ × 3³ = 6 × 6 × 6, so its cube root is 6.

A common factor divides two (or more) numbers exactly. The highest common factor is the biggest one. For small numbers, listing factors works: the factors of 18 are 1, 2, 3, 6, 9, 18 and the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 — the largest shared entry is 6.

Prime factorisation is faster for big numbers: write each number in index notation, keep only the primes that appear in both, and take the lowest power of each.

Two numbers with HCF 1, like 8 and 15, are coprime — no prime is shared, even though each number itself may be composite.

A multiple of a number is its times table: multiples of 6 are 6, 12, 18, 24, … The lowest common multiple of two numbers is the first entry they share. Listing works for small numbers: multiples of 4 are 4, 8, 12, 16, … and multiples of 6 are 6, 12, 18, … so LCM(4, 6) = 12.

For the factorisation method, use every prime that appears in either number, at its highest power: LCM(84, 126) = 2² × 3² × 7 = 252, because 84 = 2² × 3 × 7 supplies the 2² and 126 = 2 × 3² × 7 supplies the 3².

Sanity-check the size: the LCM of 8 and 12 is 24, which must be at least as large as both numbers and a multiple of each. If your LCM is smaller than one of the numbers, you found a common factor by mistake.

After solving, interpret the answer with units and sense: a piece length in cm, a coincidence time on the clock, a number of pieces. Examiners often award a final mark for the interpretation, not just the HCF or LCM.