Foundations — LCM — prime factorization method, relationship HCF × LCM = product

Before you can read a single line of the LCM page, you need to own every symbol it throws at you. Below, each one is built from nothing — plain words, then a picture, then the reason the topic can't live without it.

1. What is a "multiple"?

Look at the top row of the figure below. Starting at $0$, every jump of $4$ lands on a green dot. Every jump of $6$ lands on a coral dot. The very first place a green dot and a coral dot stack on top of each other is $12$ — the first shared landing spot.

2. What is a "factor" (divisor)?

3. The symbol $÷$ and the fraction bar $\frac{a}{b}$

The parent page writes $36/12=3$ to check that $36$ is a multiple of $12$. The check is: does $12$ fit into $36$ a whole number of times? Yes, $3$ times — so $36$ really is a common multiple.

4. What is a "prime number"?

See the middle of the figure: a prime like $5$ is a single solid block — you cannot break it into a rectangle of smaller equal rows. But $6$ can become a $2\times 3$ rectangle, so $6$ is not prime.

5. "Prime factorization" — the recipe of a number

That word unique is a promise made by a big theorem — the Fundamental Theorem of Arithmetic: every whole number above $1$ has exactly one prime recipe. No number has two different recipes. That is why comparing recipes is a safe, honest way to compare numbers.

The figure shows a factor tree for $12$: keep splitting until every branch-tip is prime. The tips read $2,2,3$ — that's the recipe.

6. Exponent notation: $2^2$, $3^2$, $5^0$

7. The dot $\cdot$ and $\times$ — both mean "multiply"

8. Letters as stand-ins: $a$, $b$, $p$, $\alpha$, $\beta$

9. The big multiply sign $\prod$ (capital Pi)

Think of it as an instruction: "go through every prime, raise it to its pile-size, and multiply the whole lot." That is exactly how you rebuild a number from its recipe.

10. $\max$ and $\min$ — pick the bigger / smaller

The figure below shows two piles of the same prime (one from $a$, one from $b$). For the LCM you keep the taller pile (max) — so both numbers still fit inside. For the HCF you keep the shorter pile (min) — that's all they truly share.

11. The "$\ge$" and "$\le$" signs

Why they show up: to be a multiple of $a$, a number must contain at least $\alpha$ copies of each prime ($\ge$). To be a divisor of $a$, it may contain at most $\alpha$ copies ($\le$). That single difference — $\ge$ for multiples, $\le$ for divisors — is why LCM uses max and HCF uses min.

12. The box   ​ and the checkmark ✓

Prerequisite map

Equipment checklist

Test yourself — you should answer each instantly before moving to the parent page.

Connections

• Parent: LCM method & HCF×LCM
• Prime Numbers and Factorization
• Fundamental Theorem of Arithmetic
• HCF — prime factorization and Euclid's algorithm
• Divisibility Rules
• Applications — bells ringing, tiling, syncing cycles