Factors, Multiples & Primes

Two questions you already ask

Hot dogs come in packs of $10$, buns in packs of $8$. How many of each pack do you buy so that nothing is left over? Different day, different problem: you have $24$ cookies and $36$ candies to split into identical gift bags with none left behind — what’s the biggest number of bags you can make? Both questions are about seeing inside whole numbers: when they split evenly, and when two counting patterns line up. In F1 you took an expression apart operation by operation; this unit takes a single number apart into its multiplicative building blocks.

One fact, seen two ways

Everything grows out of a single multiplication fact. Take $3 \times 4 = 12$. Read from one end: $3$ and $4$ are factors of $12$ — they divide it evenly, no remainder. Read from the other: $12$ is a multiple of both $3$ and $4$ — it’s a stop on their skip-counting lists. Same fact, two directions of view. Factor questions look down into a number; multiple questions look up and outward from it. Keeping that compass straight is half of this unit.

A factor pair has a shape you can see: it’s a way to arrange that many dots into a full rectangle — no gaps, no leftovers.

Before you look too closely, predict: how many rectangles can $12$ make? (Count $1\times12$, $2\times6$, $3\times4$.) Now type $7$ — nothing but the single line exists. Then $36$: it has a rectangle none of the others had, the perfect square $6 \times 6$ — which is why $36$ has an odd number of factors while almost every number has an even count (factors come in pairs, unless one pairs with itself).

Multiples are the opposite direction: start at the number and keep adding it, forever.

Light up $3$ and watch the stripes. Then try $4$, remember the pattern, and switch to $6$: the squares lit both times — $12$, $24$, $36$ — are the common multiples of $4$ and $6$, and the first one, $12$, will matter in a minute.

Primes: the atoms of multiplication

Some numbers refuse to make any rectangle except the single line. Their only factors are $1$ and themselves: $2, 3, 5, 7, 11, \dots$ These are the primes, and they play the role atoms play in chemistry — every whole number from $2$ up is built by multiplying primes, and (this is the remarkable part) in exactly one way. $60$ is $2 \times 2 \times 3 \times 5$ no matter how you find the pieces. A number with more than one rectangle is composite: it can be broken down further.

Two edge cases earn their reputation. $2$ is the only even prime — every other even number already contains $2$ as an extra factor. And $1$ is neither prime nor composite: if we called $1$ a prime, the “exactly one way” promise would collapse, since $6 = 2 \times 3 = 1 \times 2 \times 3 = 1 \times 1 \times 2 \times 3 \dots$ Mathematicians shut that door by definition.

To find a number’s atoms, split it any way you can and keep splitting until only primes remain — a factor tree:

Try $60$ and check it matches the steps above. Then predict $64$‘s leaves before typing it — how many $2$s? Then try $97$: the tree refuses to branch at all.

Divisibility shortcuts — and why they work

Factor trees need a way to spot factors quickly, and that’s what these shortcuts are for — a factor at a glance, no long division.

The digit-sum rule deserves a second look, because it feels like magic. Every $10$ is a $9$ plus $1$, every $100$ is a $99$ plus $1$ — so $342 = 3(99+1) + 4(9+1) + 2$, which regroups into (a pile of nines) $+\ 3 + 4 + 2$. The pile of nines is certainly divisible by $3$, so only the digit sum $9$ decides. That’s not a coincidence to memorize; it’s the base-ten system showing its seams.

Before checking $342$: is it even? Does $3$ go in? Does $4$? Make all three calls, then look. Then try $51$ — it feels prime, but the digit sum gives it away.

GCF & LCM — two questions, two directions

Now the opening problems. The gift bags ask: what’s the biggest number that divides both $24$ and $36$? — the Greatest Common Factor. The hot dogs ask: what’s the first number both $10$ and $8$ count up to? — the Least Common Multiple ($40$: four packs of hot dogs, five of buns).

Why do people swap them? Because both are “a number the two numbers share,” and the names are a mouthful — so the mind grabs whichever word surfaces first. The rescue is the direction compass: a common factor fits inside both numbers, so the GCF can never exceed the smaller one. A common multiple contains both numbers, so the LCM can never be below the larger one. If your “GCF of $24$ and $36$” comes out as $72$, the size alone says you answered the other question.

The prime-atom picture makes both mechanical: the GCF is the atoms the two numbers share (lowest power of each shared prime), and the LCM is the smallest collection containing each number’s full set (highest power of every prime that appears).

Set $12$ and $18$ and predict both answers first ($12 = 2^2 \times 3$, $18 = 2 \times 3^2$ — shared atoms $2 \times 3$). Then try $8$ and $9$: no shared primes at all, so the GCF drops to $1$ and the LCM has no choice but to be the full product, $72$.

The one thing to remember

Whole numbers have insides, and primes are the atoms: every number is one unique multiplication of primes. Factor questions look down into a number (GCF = the shared atoms); multiple questions look up from it (LCM = the smallest pile of atoms containing both). When you’re unsure which one a problem wants, check the direction: splitting evenly looks down, lining up looks up.

The vocabulary

• Factor — divides evenly. Factors of $12$: $1, 2, 3, 4, 6, 12$.
• Multiple — what you get by multiplying. Multiples of $12$: $12, 24, 36, \dots$
• Prime — exactly two factors, $1$ and itself: $2, 3, 5, 7, 11, \dots$   ($2$ is the only even prime; $1$ is not prime.)

Divisibility shortcuts (memorize these)

GCF vs LCM

For $60$ and $18$: