Finding Common Ground

A factor is a number that divides another number exactly without leaving a remainder (e.g., 4 is a factor of 12). A multiple is the result of multiplying a number by a positive whole number (e.g., 12 is a multiple of 4). Factors are smaller than or equal to the number, while multiples are greater than or equal to the number.

The HCF (also called GCD or Greatest Common Divisor) of two or more numbers is the largest whole number that divides each of them without a remainder. For example, the common factors of 12 and 16 are 1, 2, and 4, so HCF(12, 16) = 4.

The LCM of two or more numbers is the smallest positive whole number that is a multiple of all of them. For example, common multiples of 6 and 8 are 24, 48, 72..., and the smallest is 24, so LCM(6, 8) = 24.

Write each number as a product of prime numbers. Identify the common prime factors and take each with the minimum number of times it appears in any factorisation. Multiply these together to get the HCF.

Write each number as a product of prime numbers. List every prime factor that appears in ANY of the numbers, taking each with the maximum number of times it appears in any single factorisation. Multiply them together to get the LCM.

A common factor must divide BOTH numbers. If a prime factor appears 3 times in one number but only 1 time in the other, a common factor can contain that prime at most 1 time, because dividing the second number by more than 1 of that prime would leave a fraction.

A common multiple must be divisible by BOTH numbers. If one number contains three 2s (2³ = 8), any multiple of that number must contain at least three 2s. Therefore, the LCM must have at least as many of each prime as the number with the most occurrences.

Two numbers are co-prime (or relatively prime) if their only common factor is 1, meaning HCF(a, b) = 1. Co-prime numbers do not need to be prime themselves: for example, 8 and 9 are both composite, but they are co-prime.

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. This happens because the prime factors of a and b are partitioned into the minimum occurrences (which form the HCF) and the maximum occurrences (which form the LCM), and their product contains every prime factor exactly as many times as a × b.

No! The identity HCF × LCM = Product holds ONLY for two numbers. For three numbers like 2, 4, 8: HCF = 2 and LCM = 8 (giving HCF × LCM = 16), but their product is 2 × 4 × 8 = 64.

A conjecture is a statement or claim made based on patterns or observations that appears to be true, but has not yet been proven or verified for all cases.

A counterexample is a single specific case that disproves a general claim or conjecture. For instance, comparing 96 (6 prime factors) with 121 (2 prime factors) is a counterexample to the claim that larger numbers always have longer prime factorisations.

The HCF of any two consecutive even numbers (like 2k and 2k+2) is always 2. Their difference is 2, so any common divisor must divide 2. Since both are even, 2 is always their greatest common factor.

The HCF of any two consecutive odd numbers (like 2k+1 and 2k+3) is always 1. Their difference is 2, but neither number is divisible by 2, so their only common divisor is 1.

If one number divides the other, HCF(n, kn) = n. For example, HCF(6, 18) = 6 and HCF(7, 35) = 7.

Use HCF when a problem asks to divide, cut, or pack items into equal-sized pieces, groups, or bags of the LARGEST possible size without any leftover (e.g., finding the largest square tiles for a room, or largest equal rice bags).

Use LCM when events, cycles, or measurements repeat at different intervals and you need to find when they will NEXT happen together, or the SMALLEST common length/quantity that can be formed (e.g., when two sirens sound together, or the shortest toran made of equal strips).

Instead of dividing by prime numbers one by one (like 2, then 5, then 5), you can directly divide by any larger common factor you immediately recognize (like 50 or 10). This saves steps while producing the exact same HCF and LCM.

Every factor of a number is formed by taking some subset ('subpart') of the prime factors. For 225 = 3² × 5², we choose 3 with power 0, 1, or 2 (3 choices) and 5 with power 0, 1, or 2 (3 choices), giving 3 × 3 = 9 factors: 1, 3, 5, 9, 15, 25, 45, 75, 225.

It represents every number from 1 to 100 as a circular ring divided into equal angular segments corresponding to its prime factors: Green for 2, Purple for 3, Orange for 5, Blue for 7, and Red for primes ≥ 11. Primes appear as solid rings, while composite numbers show their multi-colored prime slices.