Mathematics · Numbers and calculation · Grade 7

Prime numbers, HCF and LCM

Two buses leave at 12 and 18 minutes — when will they leave together again? What size of tile?

The situation

From the Chilonzor stop, bus 12 leaves every 12 minutes and bus 18 every 18 minutes; at 8:00 both left together. When will they next leave together? A kitchen floor measures 18 × 12 dm; we want to cover it exactly with the largest possible identical square tiles — what will the tile be like? Both questions are answered by the divisors and multiples of a number, and their “bricks” are the prime numbers.

Topic

Prime numbers, prime factorisation, HCF and LCM. Two buses leave the same stop every 12 and 18 minutes — when will they leave together again? The sieve of Eratosthenes, the factor tree, HCF with tiles

Key ideas

Prime number
HCF
LCM

Formulas

360 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 = 2^{3} \cdot 3^{2} \cdot 5

Every natural number (except $1$ ) can be broken into prime factors in exactly one way; repeated factors are written with a power.

HCF (18; 12) = 6, LCM (12; 18) = 36

The highest common factor is the largest number that divides both numbers; the lowest common multiple is the smallest number divisible by both.

12 = 2^{2} \cdot 3, 18 = 2 \cdot 3^{2} \Rightarrow HCF = 2 \cdot 3, LCM = 2^{2} \cdot 3^{2}

Table method: the HCF is the smallest powers of the common prime factors, the LCM is the largest powers of all the factors.

HCF (a; b) \cdot LCM (a; b) = a \cdot b

For two numbers, the product of the HCF and LCM equals the product of the numbers themselves: $6 \cdot 36 = 12 \cdot 18$ .

Theory

Divisors and multiples

$18 : 6 = 3$ with no remainder — so $6$ is a divisor of $18$ , and $18$ is a multiple of $6$ . The divisors of $18$ are $1, 2, 3, 6, 9, 18$ — there are finitely many; its multiples $18, 36, 54, \dots$ are infinite. $1$ is a divisor of every number, and every number is both a divisor and a multiple of itself. Divisibility tests help to find a factor quickly: a number ending in an even digit is divisible by $2$ , one whose digit sum is divisible by $3$ is divisible by $3$ , and one ending in $0$ or $5$ is divisible by $5$ .

Prime and composite numbers

A number with exactly two divisors ( $1$ and itself) is a prime number: $2, 3, 5, 7, 11, \dots$ ; a number with more than two divisors is a composite number. $1$ is neither prime nor composite (it has one divisor). Sieve of Eratosthenes: we leave $2$ and cross out its multiples, then those of $3$ , $5$ , $7$ … Up to $100$ it is enough to stop at $7$ , because $11^{2} = 121 > 100$ : if a number still not crossed out had a smaller divisor, that divisor would not exceed $\sqrt{100} = 10$ . $2$ is the only even prime; there are infinitely many primes.

Prime factorisation

We split a composite number into two factors and split the composite ones again — the leaves of the factor tree are primes: $360 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5$ . However we start ( $360 = 36 \cdot 10$ or $4 \cdot 90$ ), the same set of leaves comes out — this is the fundamental theorem of arithmetic. Repeated factors are written compactly with a power: $2^{3} \cdot 3^{2} \cdot 5$ (the “Powers” experiment). The ladder method does the same: we keep dividing the number by its smallest prime divisor.

HCF — highest common factor

The common divisors of $18$ and $12$ are $1, 2, 3, 6$ ; the largest is $6$ — $HCF (18; 12) = 6$ . In the picture: the largest square tile that covers an $18 \times 12$ floor exactly is $6 \times 6$ . Euclid's algorithm: divide the larger number by the smaller with a remainder, then divide the divisor by the remainder — until the remainder is zero; the last non-zero remainder is the HCF: $18 = 12 \cdot 1 + 6$ , $12 = 6 \cdot 2 + 0$ . If the HCF $= 1$ , the numbers are coprime ( $8$ and $15$ ). The HCF cancels a fraction: $\frac{12}{18} = \frac{2}{3}$ .

LCM — lowest common multiple

The multiples of $12$ are $12, 24, 36, 48, \dots$ ; those of $18$ are $18, 36, 54, \dots$ ; the first common one is $36$ — $LCM (12; 18) = 36$ : the buses leave together again after $36$ minutes. By the table method: $12 = 2^{2} \cdot 3$ , $18 = 2 \cdot 3^{2}$ ; for the HCF we take the smaller powers of the common factors ( $2 \cdot 3 = 6$ ), for the LCM the larger powers of all of them ( $2^{2} \cdot 3^{2} = 36$ ). Hence $HCF \cdot LCM = a \cdot b$ . The LCM is the common denominator of fractions: $\frac{1}{12} + \frac{1}{18} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36}$ .

About the model

The sieve is shown in the drawing for numbers up to $200$ , the tree up to $999$ — the methods themselves work for any number, only for large numbers the division steps multiply. The tile side is in whole decimetres; the bus timetable is taken as ideal (no delays). The chapter's main points: divisor, multiple, prime and composite numbers, prime factorisation, HCF, LCM and their use with fractions — all are shown in the steps.

Did you know?

Eratosthenes (about 276–194 BC) headed the Library of Alexandria: he devised the method of “sieving” the prime numbers and was the first to calculate the circumference of the Earth.
There are infinitely many primes — Euclid proved this in Book IX of the “Elements” (about 300 BC). The largest known prime, found in October 2024, is $2^{136 279 841} - 1$ — it has more than 41 million digits.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.

More experiments in this section

Fractions: from pizza to formula — Grade 5
The number line: from −7 to √2 — Grade 7
Powers: from folding to a formula — Grade 7
Roots: from squares to powers — Grade 8