Prime Time

Why This Matters

Some numbers are very friendly. You can split them into equal groups many ways. Take 12. You can make 2 groups of 6. Or 3 groups of 4. Or 4 groups of 3. Lots of choices.

But some numbers are stubborn. Take 7. Try to share 7 toffees equally among friends. You can give 1 toffee each to 7 friends, or all 7 to 1 friend. That is all. No other neat way works.

These stubborn numbers are called prime numbers. They are the building blocks of all other numbers. Every number is made by multiplying primes together, just like every word is made from letters.

In this chapter you will learn to find a number’s building blocks. You will play two fun games to spot factors and multiples. You will meet co-prime numbers. And you will learn quick tricks to check if a big number can be divided by 2, 3, 4, 5, 6, 9 or 10 — without doing long division. These tricks will save you time for the rest of your life.

The Big Idea

Every counting number bigger than 1 is built by multiplying prime numbers together. A prime is a number with exactly two factors — 1 and itself. Primes are like the bricks of the number world. Once you know which bricks build a number, lots of things become easy: you can tell what divides it, whether two numbers share a common factor, and how two numbers line up. This chapter is about finding those bricks.

Let’s Break It Down

Factors and Multiples

Two small words do a lot of work in this chapter: factor and multiple. Let us meet them.

A factor of a number is a number that divides it exactly. “Exactly” means nothing is left over — the remainder is 0.

For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. Each one divides 12 with no remainder. Check: 12 ÷ 3 = 4, no remainder. So 3 is a factor.

Here is a nice way to see the factors of a number. Imagine 12 figs (anjeer). Try to line them up in neat rows of equal size. Each neat rectangle gives you a factor pair.

The picture below shows every way to arrange 12 figs in equal rows.

Tap to zoom

Now the other word. A multiple of a number is what you get when you multiply that number by 1, 2, 3, 4, and so on.

For example, the multiples of 3 are 3, 6, 9, 12, 15, 18, … (that is 3 × 1, 3 × 2, 3 × 3, …). The list never ends.

These two words are two sides of the same coin. If 3 is a factor of 12, then 12 is a multiple of 3. They always come in a pair.

Multiplication tables are the key to both ideas, so let us refresh them.

Let us find every factor of a number by checking, one by one, what divides it.

Common Factors and Common Multiples (the Idli-Vada game)

Sometimes we want to compare two numbers. We look for the factors they share, and the multiples they share.

There is a lovely game in your textbook for this. It is called the Idli-Vada game.

Children sit in a circle and count: 1, 2, 3, 4, … But there are two rules:

• When your number is a multiple of 3 (like 3, 6, 9, …), you say “idli” instead of the number.
• When your number is a multiple of 5 (like 5, 10, 15, …), you say “vada” instead.
• And when your number is a multiple of both 3 and 5, you say “idli-vada”!

So when do you say “idli-vada”? Only for numbers that are in both tables. The first such number is 15 (it is 3 × 5). The next is 30. These shared multiples are called common multiples of 3 and 5.

The picture below shows the game played from 1 to 30. Two circles overlap, and the overlap holds the idli-vada numbers.

Tap to zoom

A common multiple of two numbers is a number found in both their tables. A common factor of two numbers is a number that divides both of them exactly.

Let us find common factors with a worked example.

Prime and Composite Numbers (and why 1 is neither)

Now look again at how numbers behave when you try to arrange them in neat rows.

Some numbers, like 7, can be arranged in only one way: a single row. Their only factors are 1 and the number itself.

Other numbers, like 8 or 12, can be arranged in more than one way. They have extra factors hiding inside.

The picture below shows this difference clearly.

Tap to zoom

So here are the two big words:

A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13.

A composite number has more than two factors. Examples: 4, 6, 8, 9, 10, 12.

A clever old way to find all the primes is the Sieve of Eratosthenes (named after a Greek mathematician from long ago). You write the numbers in a grid, cross out 1, then keep one of each prime and cross out all its bigger multiples. Whatever stays uncrossed is prime.

The figure below shows the result up to 30.

Tap to zoom

Now, the special case. Why is 1 neither prime nor composite?

This is not just a rule to memorise. There is a clear reason.

Count the factors of 1. The only number that divides 1 is 1 itself. So 1 has just one factor.

• A prime needs exactly two factors. 1 has only one, so 1 is not prime.
• A composite needs more than two factors. 1 has only one, so 1 is not composite.

1 falls into neither box. So we say 1 is special — it is neither prime nor composite. That is why the grey box for 1 stands alone in Figure 5.4.

Prime Factorisation

Primes are the building bricks. So we can take any composite number and break it down until only primes are left. This breaking-down is called prime factorisation.

The easiest way to do it is a factor tree. You split the number into any two factors. Then split those factors. Keep splitting until every branch ends in a prime. A prime cannot split any more, so it becomes a “leaf” of the tree.

Let us build a factor tree for 36, step by step.

The figure below shows this whole tree in one picture.

Tap to zoom

Here is something wonderful. You can split a number in different ways at the start. You could have begun with 36 = 6 × 6, or 36 = 2 × 18. But you always end up with the same set of primes: two 2s and two 3s. The order may change, but the bricks never do. Every number has only one prime recipe.

Co-prime numbers

Sometimes two numbers share no factors at all — except the 1 that everybody shares. Such a pair has a special name: co-prime.

Two numbers are co-prime if their only common factor is 1.

Be careful: the numbers themselves do not have to be prime! For example, 8 and 9 are both composite. But check their factors:

• Factors of 8: 1, 2, 4, 8
• Factors of 9: 1, 3, 9

The only number in both lists is 1. So 8 and 9 are co-prime, even though neither is prime.

Prime factorisation gives a quick way to test this: if two numbers share no common prime brick, they are co-prime.

Divisibility rules (and the simple reason for each)

Imagine a big number like 8560. Is it divisible by 2? By 5? By 4? Doing long division each time is slow. Luckily there are quick tricks. They are called divisibility rules.

The figure below is a handy table. We will explain why each one works, because none of them is magic.

Tap to zoom

Let us see why each rule is true.

By 10 — last digit is 0. Our number system counts in tens. Every multiple of 10 (10, 20, 30, …) ends in 0. So a number splits neatly into tens only when it ends in 0.

By 5 — last digit is 0 or 5. Count in fives: 5, 10, 15, 20, 25, … The last digit is always 0 or 5. So those are the only endings that work.

By 2 — last digit is even (0, 2, 4, 6, 8). Every group of 10 splits into 2 evenly (10 = 2 × 5). So the tens part of any number is always divisible by 2. Only the last digit decides. If the last digit is even, the whole number is.

By 4 — the last two digits make a multiple of 4. Here 100 is the key: 100 = 4 × 25, so any number of hundreds divides evenly by 4. Thousands, hundreds — all gone. Only the last two digits are left to decide. For 8536, just check 36. Since 36 = 4 × 9, the whole of 8536 is divisible by 4.

By 3 — the digits add up to a multiple of 3. Here is the gentle reason. 9 is divisible by 3, and so is 99, and 999. So if you take away all the full tens, hundreds and thousands (which are made of 9s, 99s, 999s plus a little), what is left over is exactly the sum of the digits. So the whole number divides by 3 only when the digit sum does. Example: 8560 → 8 + 5 + 6 + 0 = 19. 19 is not a multiple of 3, so 8560 is not divisible by 3.

By 9 — the digits add up to a multiple of 9. Same idea, but with 9 instead of 3 (because 9, 99, 999 are all multiples of 9). Example: 8550 → 8 + 5 + 5 + 0 = 18, and 18 = 9 × 2. So 8550 is divisible by 9.

By 6 — divisible by both 2 and 3. This one builds on the others. 6 = 2 × 3. So a number divides by 6 only when it divides by 2 and by 3 at the same time. Example: 8550 ends in 0 (so divisible by 2), and its digits add to 18 (so divisible by 3). Both true, so 8550 is divisible by 6.

Common Mistakes

Some ideas in this chapter trick almost everyone at first. Here are the traps to watch for.

Quick Check

Time to test yourself. Pick one answer in each.

Practice Problems

Try each one yourself first. Then tap to check the full solution.

Easy

Medium

Challenge

Summary

• A factor of a number divides it exactly (no remainder). A multiple is the number times 1, 2, 3, … If 3 is a factor of 12, then 12 is a multiple of 3.
• A common factor divides both numbers; a common multiple appears in both tables. The Idli-Vada game finds common multiples.
• A prime number has exactly two factors (1 and itself). A composite number has more than two.
• The number 1 is special — it has only one factor, so it is neither prime nor composite.
• Prime factorisation breaks a number into its prime bricks, using a factor tree. Every number has only one set of prime bricks. Example: 36 = 2² × 3².
• Two numbers are co-prime if their only common factor is 1. They need not be prime themselves (8 and 9 are co-prime).
• Divisibility rules let you check division quickly: by 2 (even last digit), 5 (ends in 0 or 5), 10 (ends in 0), 4 (last two digits), 3 and 9 (digit sum), and 6 (divisible by both 2 and 3).

What’s Next

You now know how numbers are built and split. Next comes shape and measurement. In Chapter 6 — Perimeter and Area, you will learn to measure the distance around a shape (its perimeter) and the space it covers (its area). The neat news: the multiplication and factor skills you just practised will help you there too, especially when working with rectangles.