A signal from memory

Erklärus picked up the radio signal of a real pulsar – a star that flashes at extremely regular intervals. But this signal was different: it pulsed at intervals of 2, 3, 5, 7, 11… “These numbers seem familiar! Our beacon at home sent like that too – but I no longer understand the pattern exactly.”

Erklärus: “Those are prime numbers – numbers with a very special property. Let’s first understand divisibility, then you’ll recognise the pattern of your signal.”

✖️ Multiples – the times table of a number

The multiples of a number are all the results you get when you multiply it by 1, 2, 3, 4…

Multiples of 4

M₄ = {4, 8, 12, 16, 20, 24 …}

Is 18 a multiple of 6? (yes=1, no=2)

➗ Factors – the numbers that fit in exactly

The factors of a number are all the numbers that you can divide it by without a remainder.

Factors of 12

F₁₂ = {1, 2, 3, 4, 6, 12}

Erklärus: “The best trick when looking for factors: always find pairs that multiply to give the number.”
1 × 24
2 × 12
3 × 8
4 × 6

This is how you are sure to find all the factors of 24 without forgetting one.

Is 5 a factor of 20? (yes=1, no=2)

🧠 Divisibility rules – the quick tricks

These rules tell you straight away whether a number is divisible – without calculating at all:

Divisible by 2, 5, 10

By 2: ends in 0, 2, 4, 6, 8. By 5: ends in 0 or 5. By 10: ends in 0.

Divisible by 3 – the digit sum

Add up all the digits. If the result is divisible by 3, then the whole number is too!

153: 1+5+3 = 9 → 9 is divisible by 3 → so is 153!

Digit sum of 246?

💎 Prime numbers – the building blocks of all numbers

A prime number has exactly two factors: itself and 1. The number 1 itself is not a prime number!

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

Gold = prime number · Red = the 1 (not a prime, only one factor)

Liri: “The 2 is the only even prime number – all the other even numbers have 2 as a factor too!”
Erklärus: “Well spotted. Prime numbers are the building blocks from which all other numbers can be put together by multiplication.”
Is 17 a prime number? (yes=1, no=2)
🌌 Prime numbers in the universe

Some species of cicada on Earth live in cycles of 13 or 17 years – both prime numbers. Researchers suspect this protects them from predators with other life cycles.
Prime numbers are also used to send messages into space – they can be recognised universally, no matter what language a receiver speaks.

❌ Common mistake

The 1 is often wrongly thought to be a prime number! But it has only a single factor (itself) – a prime number needs exactly two.

📝 Exercise 1
Name the biggest factor of 15 that is not 15 itself.
📝 Exercise 2
Is 402 divisible by 3? (yes=1, no=2)
📝 Exercise 3
Which prime number comes straight after 7?

The signal decoded

Liri checked her remembered sequence of numbers once more: 2, 3, 5, 7, 11 – all prime numbers! She wrote the beacon pattern onto her star map as a recognition signal.

Liri: “It was a prime-number signal all along! That is how we sent messages into space at home, because prime numbers can be recognised the same everywhere.”
Erklärus: “A clever choice by your people – and now you know exactly what to look for in the sky. Every memory you decode brings your map closer to the truth.”
“Multiples: results of the times table. Factors: divide without a remainder, look for factor pairs. Divisibility rules: 2/5/10 by the last digit, 3 by the digit sum. Prime number: exactly 2 factors, 1 doesn’t count.”
What you learned today
F₁₂ = {1,2,3,4,6,12}
Look for factor pairs, check the digit sum, recognise prime numbers
🌟

Chapter 26 mastered!

Liri can now confidently recognise factors, multiples and prime numbers.

Erklärus’s notebook – page 26:
“Look for factor pairs so as not to forget any. Digit sum for the rule of 3. Prime numbers have exactly two factors – the 1 doesn’t count.”
← Back to overview 26 – Divisibility & Primes