PrepShorts · Study sheet · Class 10 Mathematics · Chapter 1, Real Numbers
Chapter 1 · Real Numbers
Why a composite number has one prime factorisation and no other
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32760 opens into primes 47 different ways at the top; chase every one to the bottom, and 15015 distinct factor trees come out, every single one ending at the same eight primes.
The idea
That a composite number breaks into primes at all is the cheap half of Theorem 1.1; you can see it happen on any factor tree. The half that earns the theorem its name is that the same primes come out every time, because that is what licenses a negative claim — if a prime never shows up in one factorisation, it is absent from every factorisation, so you may state flatly that the number is not divisible by it. Example 1 is exactly that shape, and so is Exercise 1.1 Q5 — neither would be safe if a second, different factorisation were lurking somewhere. The chapter leans on uniqueness the other way round too, and an explanation should show both directions rather than only the denial: the prime factorisation route to HCF and LCM named on p. 4 builds a number up out of indices, and the proof of Theorem 1.2 on p. 6 twice concludes that a prime is one of a listed set. Uniqueness is what makes a factorisation something you can read facts off at all; ruling divisibility out is the most striking of those facts, not the only one.
What you should be able to do
- Generate composite numbers by multiplying a chosen set of primes with repetition, and explain why that process alone never proves the reverse direction
- Build the factor tree of a large number and collect its leaves into a product of prime powers
- Build a second, differently branched tree for the same number and observe that the collection of leaves is unchanged
- State what Theorem 1.1 asserts, separating the existence claim from the uniqueness claim, and say which of the two is doing the work in a given argument
- Write a factorisation in the chapter's ascending-prime, combined-powers form, and explain why fixing the order removes the theorem's only escape clause
- Argue that a stated number cannot end in the digit zero by showing that the prime 5 is missing from its factorisation
- Explain why an expression such as a product of primes plus a shared factor is composite, without computing the whole value first
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| prime | a whole number above 1 whose only whole-number divisors are itself and 1 | printed throughout §1.2, p. 2 |
| composite number | a whole number above 1 that is not prime, so it has a divisor other than itself and 1 | printed in §1.2, p. 2 |
| Fundamental Theorem of Arithmetic | the result that a composite number splits into primes, and that the collection of primes it splits into is fixed | printed as the heading of §1.2 and as Theorem 1.1, pp. 2–3 |
| prime factorisation | the way a number is written once every factor in it has been reduced to primes | printed in §1.2, p. 3 |
| factor tree | the branching diagram that splits a number, then splits each factor again, until only primes are left at the tips | printed in §1.2, p. 2, with the tree itself drawn as artwork |
| powers of primes | the same factorisation with each repeated prime collected under an index | printed in §1.2, p. 3 |
| conjecture | a statement believed true from examples but not yet proved | printed in §1.2, p. 3, describing the stage before Theorem 1.1 is stated |
| ascending order | the convention of listing the primes of a factorisation smallest first | printed in §1.2, p. 4 |
| natural number | a counting number, 1, 2, 3, and onwards | printed in §1.2, p. 2 |
| canonical form | one agreed way of writing a factorisation so that two of them can be compared symbol by symbol | an added term; the chapter fixes such a form on p. 4 without naming it |
| multiset | a collection in which repeats count, so three 2s differ from two 2s | an added vocabulary; not printed in this chapter |
Where people slip up
- "The theorem just says numbers can be factorised." Half of it does, and that half is visible on any tree. The clause that matters says there is no second, disagreeing factorisation hiding anywhere. Split the statement into its two claims and keep them apart.
- "Different factor trees give different answers." Students who start 32760 at 360 × 91 rather than 2 × 16380 often expect a different result and are surprised into thinking they made an error. Run both trees side by side so the agreement is something they watched happen.
- "3803 × 3607 is a prime factorisation." Not until someone has checked that neither factor splits further. The chapter is unusually explicit about handing this check to the reader.
- "A factorisation is unique, full stop." Only once order is pinned down. 2 × 3 × 5 × 7 and 7 × 5 × 3 × 2 are two different strings and one factorisation, which is exactly why the chapter fixes ascending order.
- "4ⁿ never ends in 0 because I checked the first six powers." A pattern in a table is a reason to suspect, not a reason to assert. The proof is that the prime 5 is not available, and it covers every n at once.
- "1 is prime, so it can go in the factorisation." Allowing 1 would let you pad any factorisation with as many extra factors as you liked, and uniqueness would die immediately. That is the structural reason 1 is excluded.
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Worked answers: Exercise 1.1 · Exercise 1.2 · this video explains Exercise 1.1 Q1, Exercise 1.1 Q5, Exercise 1.1 Q6
Transcript1,736 words
Here are two things you can do with whole numbers, and they are not the same difficulty. You can take a handful of primes and multiply them together. Nothing can go wrong in that direction. Pick your primes, multiply, and out comes a number. Or you can start with a number and pull it apart into primes. That is the direction where all the interest is, and this video asks one question about it.
Not whether it can be done. That turns out to be the cheap half. The question is whether it can be done in more than one way. Whether two people, taking the same number apart, following different hunches, could end up holding different primes. The answer is no, and that no is doing an enormous amount of work. Start with the easy direction, and be systematic about it. Here is a supply of primes: 2, 3, 7, 11 and 23.
Take some of them, in any quantity you like, and multiply. 7 times 11 times 23 gives 1771. Put a 3 in front of that and you get 5313. Put a 2 in front of that and you get 10626. You are not obliged to use them all, and you may use one more than once. 2 cubed, times 3, times 7 cubed is 8232, which touches only three of the five and repeats two of them.
That is what the word supply is doing. It is not a checklist. And every number this machine produces arrives already factorised, because you built it out of its own factors. Now turn the machine round. Hand it a number instead of primes, and ask it to give the primes back. Ask the small question first. Is there a composite number sitting somewhere that no such product can reach? There is not, and you can see why without much trouble.
Take any composite number. By what composite means, it splits into two smaller factors. If either of those is composite, split it again. The pieces only ever get smaller, so the splitting cannot go on forever, and it stops exactly when everything you are holding is prime. So every composite number does break into primes. That is settled, and it was not hard. The hard question is the one lying underneath it.
Take 32760 and do it. 32760 is even, so pull out a 2, and 16380 is left. That is even too. Another 2, and 8190. Even again. A third 2, and 4095. 4095 is odd now, so the run of 2s stops. But its digits add to 18, and 3 divides 18, so 3 comes out. 1365 is left. The digits of 1365 add to 15, so another 3 comes out, leaving 455.
455 ends in 5, so out comes a 5, and 91 is left. And 91 is 7 times 13. Stop. Everything at the tips is prime. Read the tips. 2, 2, 2, 3, 3, 5, 7 and 13. Eight primes. Now do it again, and deliberately do it differently. Nothing forced those first moves. Taking the smallest prime each time was a habit, not a rule. So this time notice that 32760 is 360 times 91, and start there instead.
360 is 8 times 45. 8 is 2 times 4, and 4 is 2 times 2. 45 is 9 times 5, and 9 is 3 times 3. And 91, as before, is 7 times 13. A different first move, and a different route down. Of the seven numbers this tree stops at on the way, five never appeared in the first tree at all. Read the tips. 2, 2, 2, 3, 3, 5, 7 and 13.
The same eight primes, by a different road. Two trees agreeing is a pleasant surprise. It is not evidence of very much. A student is entitled to ask whether some third tree, one nobody happened to draw, disagrees. So do not stop at two. 32760 can be opened at the top in 47 different ways. 47 pairs of factors, both above 1, that multiply back to it. And each piece can then be opened in every way it allows, all the way down.
Count that whole family and there are 15015 different factor trees for this one number. Fifteen thousand and fifteen ways to take it apart. So build every one of them, and read the tips of every one of them. Every single tree, all 15015, ends in 2, 2, 2, 3, 3, 5, 7 and 13. Not most of them. Not the ones you would naturally draw. All of them. That is no longer a surprise. That is a measurement.
Write down what has just been claimed, because it is two claims wearing one coat. First. Every composite number can be written as a product of primes. Second. It can be written that way in only one manner. The first is the one you can watch happen on any tree, and we argued it in a single sentence. The second is the one that costs something, and it is the one with teeth.
Together they are called the Fundamental Theorem of Arithmetic, and the word fundamental is not decoration. Almost everything you will ever do with whole numbers leans on the second clause without mentioning it. So it is worth being able to say which of the two you are using at any moment. There is a loose thread here, and it has to be tied off or the theorem is simply false.
The eight primes of 32760 can be written down in 3360 different orders. Smallest first, or largest first, or the 13 in the middle. Are those 3360 different factorisations? Obviously not. They are one factorisation, written down 3360 ways. So agree to write them smallest first, always. Exactly one of the 3360 orderings is the ascending one, and that is the one everybody writes. Then collect the repeats under indices. 2 cubed, times 3 squared, times 5, times 7, times 13.
Now two factorisations of one number are two strings of symbols, and you can compare them symbol by symbol. Order was the only freedom you were ever given, and fixing it is what makes uniqueness something you can state at all. There is a second thread, and pulling it explains a rule that can look arbitrary. 1 is not a prime. You were told that, and probably not told why.
Here is why. Suppose 1 were allowed into a factorisation. 12 is 2 times 2 times 3. But it would also be 1 times 2 times 2 times 3. And 1 times 1 times 2 times 2 times 3. And so on, without end. Allow four factors and there are two factorisations. Allow five and there are three. Allow six and there are four. Uniqueness would not merely weaken. It would fail by as much as you cared to make it fail.
So 1 is kept out, and it is kept out precisely to protect this theorem. Not every number gives itself away as easily as 32760 did. Take 123456789. Its digits add to 45, and 45 is divisible by 9, so a 9 comes out. That leaves 13717421, and 13717421 is 3803 times 3607. So there it is. 3 squared, times 3803, times 3607. Except that is not yet a prime factorisation, and it matters to notice why.
Nobody has checked that 3803 and 3607 are prime. Until somebody does, that line is a factorisation into three numbers. It is not a factorisation into primes. So check. To settle a number below 3844 you need only divide by primes up to 61, because 62 squared is 3844 and clears both. There are 18 primes up to 61. Try all 18 on each of them. None divides either one. Now it is a prime factorisation.
A fair question at this point is whether any of this has actually been proved. Not here it has not. Watching 15015 trees agree is powerful, and it is not a proof. 32760 is one number, and the claim is about all of them. The existence half goes back to Euclid, and it is old. The uniqueness half is much harder than it looks, and it waited a very long time.
The first correct proof of it is due to Carl Friedrich Gauss, more than two thousand years later. That gap is the best evidence available that the second clause is not obvious. It only looks obvious because it is true, and because you have never once seen it fail. Now the payoff, and it is a negative one. Can a power of 4 ever end in the digit zero? Look at the first few. 4, 16, 64, 256, 1024, 4096.
The last digits run 4, 6, 4, 6, 4, 6. It never happens. But that is an observation, not an argument. Patterns break. Here is the argument. A number ends in zero exactly when 10 divides it, and 10 divides it exactly when both 2 and 5 do. So a power of 4 that ended in zero would have to have a 5 inside it. And 4 to the n is 2 to the 2n. Its factorisation is nothing but 2s.
Uniqueness is what says that is the only factorisation there is, so no 5 is hiding in some other one. There is no 5. So there is no zero. For every n at once, and not merely for the six we looked at. That is a move you can make whenever you like. To rule a divisor out, factorise, and look for the prime that is missing. Your turn. Can 6 to the n end in zero?
6 to the n is 2 to the n times 3 to the n. There is no 5. So it cannot. And the same habit of reading answers a question that looks quite different. Is 7 times 11 times 13, plus 13, a prime number? You do not need the total. 13 is a factor of the first term, and 13 is the second term, so 13 divides the sum.
A number with a divisor other than itself and 1 is composite, and you saw that without multiplying anything out. That is what uniqueness buys you. A factorisation stops being a calculation you performed, and becomes a thing you can read facts off. Including, most usefully of all, the facts that are not there.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Comes up again in
- Reading HCF and LCM straight off two factorisationsClass 10 · Ch 1, Real Numbers
- Why the HCF-times-LCM shortcut works for two numbers but breaks for threeClass 10 · Ch 1, Real Numbers
- A prime dividing a square must divide the number itselfClass 10 · Ch 1, Real Numbers
Either side of this one
- A GP plots as a curve, and what that curve tells youClass 9 · Ch 8, Predicting What Comes Next: Exploring Sequences and Progressions