Every whole number bigger than 1 is built from prime "bricks" in exactly one way. This companion walks through that idea, uses it to prove numbers like √2 are irrational, and gives you tools, flashcards and a quiz to test yourself.
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🧬1.1 Every number has a "DNA"
Think about water. No matter where you find it — a river, a raindrop, a swimming pool — it always breaks down into the same two ingredients: hydrogen and oxygen, H₂O. That's its molecular DNA, and it never changes.
Numbers work the same way. You already know the forward direction — multiply small primes together to build a number: 2 × 2 × 3 = 12. This chapter flips that around and asks: if I hand you any number, can you always break it back down into primes? And will everyone who tries get the same answer?
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Think of primes as LEGO bricks. A composite number (like 12, 30, 100) is any number bigger than 1 that isn't prime — it's a structure built from smaller brick pieces. A prime (2, 3, 5, 7, 11…) is a single brick that can't be broken down any further.
🔑1.2 The Fundamental Theorem of Arithmetic
Theorem 1.1. Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
In everyday words: every number bigger than 1 has exactly one true recipe made of primes. It doesn't matter who does the factorising — you, a classmate, or a computer — everyone lands on the exact same list of prime "ingredients." 2 × 3 × 5 × 7 and 7 × 5 × 3 × 2 are just the same recipe written in a different order.
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It's like a cake recipe. A chocolate cake is always flour + sugar + cocoa + eggs, whatever order you list them in. You could never accidentally make the exact same cake using a completely different set of ingredients. Numbers are the same — 60 is always 2×2×3×5, no matter how you slice the search.
🌳Worked example — the "family tree" of 32760
The easiest way to find a number's prime recipe is a factor tree — like tracing a family tree backward through ancestors until you hit the oldest generation (the primes), who have no "parents" left to divide out.
Write the primes smallest-to-largest and that's the official recipe for 32760 — guaranteed, nobody will ever find a different one.
🛠️Try it yourself in the Build tab — type any number and the factor-tree tool grows the whole tree for you instantly.
🏃1.2 HCF and LCM — the racetrack problem
Here's a classic real-world puzzle from this chapter: Sonia takes 18 minutes to drive one lap of a track, Ravi takes 12 minutes. They start together — after how many minutes will they cross the starting line together again?
That question is really asking for the LCM (Lowest Common Multiple) of 18 and 12 — the first time both their "laps" line back up. If instead you wanted the biggest chunk of time that divides evenly into both lap times, that's the HCF (Highest Common Factor).
Since every number has one fixed prime recipe, comparing two numbers' recipes instantly tells you everything about how they overlap:
🔻 HCF = multiply the smallest power of every prime the two numbers share.
🔺 LCM = multiply the largest power of every prime that shows up in either number.
Example: 6 and 20
6 = 2¹ × 3¹
20 = 2² × 5¹
HCF(6, 20) = 2¹ = 2 (smallest power of the only shared prime, 2)
LCM(6, 20) = 2² × 3¹ × 5¹ = 60 (largest power of every prime seen)
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Picture two gear wheels with 6 and 20 teeth. HCF tells you the biggest "notch size" that fits both gears evenly. LCM tells you how many teeth must pass before both gears return to their starting position together — exactly like Sonia and Ravi meeting at the start line.
Handy shortcut (works for exactly two numbers). HCF(a, b) × LCM(a, b) = a × b.
Check: HCF(6,20) × LCM(6,20) = 2 × 60 = 120 = 6 × 20 ✓
⚠️Trap: this shortcut breaks for three or more numbers. For 6, 72, 120: HCF = 6, LCM = 360, so HCF × LCM = 2160 — but 6 × 72 × 120 = 51840. Not equal! For three numbers you instead need:
👉 Head to Build to plug in any two numbers of your own and watch this get worked out live.
🎯1.2 Quick puzzle: can 4ⁿ ever end in 0?
A number ends in the digit 0 only when it's divisible by 10 = 2 × 5 — meaning 5 has to be one of its ingredients.
Now look at 4ⁿ = (2²)ⁿ = 2^(2n). Its only ingredient, ever, is 2. Because every number's recipe is locked and unique (Theorem 1.1), the prime 5 can never sneak into 4ⁿ's recipe from nowhere. So 4ⁿ can never end in 0, no matter how big n gets.
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Like a pizza with a fixed recipe. If a pizza's recipe is officially "just dough, tomato, cheese" — it can never secretly contain pineapple, no matter how many more times you bake it. That's the same logical trick we'll reuse below to prove √2 is irrational.
🪜1.3 Theorem 1.2 — the stepping stone
Theorem 1.2. Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.
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In plain terms. if a prime ingredient shows up in a number squared, it couldn't have appeared out of thin air — it must already have been an ingredient of the original number. Squaring a recipe just doubles every ingredient's amount; it never invents a brand-new one.
📐1.3 Proving √2 is irrational
Here's where it gets fun. Draw a square with side length 1 unit. Its diagonal is exactly √2 units long — you can literally draw it with a ruler. Yet √2 can never be written as a neat fraction p/q. That's what "irrational" means: no matter how hard you try, no ratio of whole numbers ever lands exactly on it.
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Real, measurable, but never a "clean" fraction. √2 ≈ 1.41421356… forever, with no repeating pattern. You can point to it on a ruler, but you can never write it as (some integer)/(some integer).
We prove this with a favourite math trick — proof by contradiction: assume the opposite is true, follow the logic honestly, and watch it collapse into nonsense. That collapse proves the opposite (our original claim) must be correct.
ASSUMESuppose, to the contrary, √2 is rational. Then √2 = a/b for coprime integers a, b (b ≠ 0) — "coprime" meaning we've already cancelled any common factor, like a fraction in its simplest form.
REARRANGEb√2 = a. Square both sides: 2b² = a². So a² is divisible by 2.
APPLY THM 1.2Since 2 is prime and 2 | a², Theorem 1.2 gives 2 | a. So write a = 2c for some integer c.
SUBSTITUTE2b² = (2c)² = 4c², so b² = 2c². This means 2 | b², and again by Theorem 1.2, 2 | b.
CONTRADICTIONNow both a and b are divisible by 2 — but we started by assuming a, b were coprime (no shared factor). That's a flat-out contradiction!
CONCLUDEOur starting assumption must have been wrong. Therefore √2 is irrational. ∎
💡 Swap every "2" in this proof for "3" and you've just proved √3 is irrational too. Swap it for any prime p, and the exact same five steps prove √p is irrational. Theorem 1.2 is the reusable engine that makes all of this work.
⚗️1.3 Mixing rational and irrational numbers
Two more useful facts, both proved the same contradiction way:
➕ The sum or difference of a rational number and an irrational number is irrational.
✖️ The product or quotient of a non-zero rational number and an irrational number is irrational.
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Like mixing paint. A "clean" rational number is like clear water; an irrational number is like a single drop of dye that never fully dissolves into a repeating pattern. Add clear water to that dye (add/subtract/multiply by a rational), and the mixture is still permanently "tinted" — it never becomes perfectly clear again.
Example: 5 − √3 is irrational
ASSUMESuppose 5 − √3 = a/b (rational, coprime a, b).
ISOLATE√3 = 5 − a/b = (5b − a)/b — a ratio of integers, i.e. rational.
CONTRADICTIONBut √3 is irrational (proved earlier). A number can't be both. So the assumption fails — 5 − √3 is irrational. ∎
Example: 3√2 is irrational
ASSUMESuppose 3√2 = a/b (coprime a, b).
ISOLATE√2 = a/(3b) — rational, since 3, a, b are all integers.
CONTRADICTIONContradicts √2 being irrational. So 3√2 is irrational. ∎
💡Pattern to remember. to prove an expression is irrational, assume it equals a/b, algebraically isolate the "known irrational" (like √2 or √3) on one side, and show that forces it to be rational — contradiction.
📋1.4 Summary
🧬 Fundamental Theorem of Arithmetic: every composite number has one fixed prime "recipe," order aside.
🪜 Theorem 1.2: if prime p divides a², then p divides a — squaring never invents a new ingredient.
📐 Irrationality: √2, √3 (and any √p for prime p) are irrational — proved by contradiction using facts 1 and 2 together.