Types of number and prime factorisation

Behind every online payment sits a very large number. It was built by multiplying two primes together. A computer did that in an instant. Going backwards is the hard part. Try to find the two primes inside it, and the best computers we have would still be working centuries from now. Easy one way, hard the other. Your numbers today are small, but the job is the same. Take a number apart and find the primes it is built from. First the words, then the method.

This is the first video in our chapter on factors, multiples and counting. The full chapter is on screen now.

Counting numbers first. One, two, three, and on forever. Those are the natural numbers. Add zero and the negatives and you have the integers. Whole numbers only, no fractions. Square and cube numbers are named after shapes. Four tiles make a two by two square. Twenty seven blocks stack into a three by three by three cube.

Quick one, and you can get this. Which of these is a cube number: eight, nine, or twelve? Have a think. I'll wait. The answer is eight. Two times two times two. Nine is three times three, so nine is a square. Twelve is neither.

Now the pair people mix up. A rational number is one you can write as an integer over an integer. Four is four over one. A half already is. Flip a number over and you get its reciprocal, so four becomes a quarter, and any number times its reciprocal makes one. Here is where students slip. A third as a decimal is nought point three three three, forever. Forever does not make it irrational. It is one over three, so it is rational. Irrational means no fraction of two integers works at all. Root two is one. Its decimal never stops and never repeats. Pi is another. So, the sort. Minus three: integer, rational. Zero: integer, rational. Four and nine: natural, integer, square, rational. Twenty seven: natural, integer, cube, rational. A half: rational. Root two: irrational.

One word left, and the rest of the video runs on it. A prime has exactly two factors: one, and itself. Seven is prime. Which tells you why one is not prime. One has exactly one factor, itself. It cannot have two. Primes start at two, and two is the only even one. Second thing. Two times fifteen is thirty. True, but fifteen is not prime. It splits again into three and five. A factor pair is not a prime factorisation.

So, thirty. Which of these is its prime factorisation? One times two times three times five. Two times fifteen. Or two times three times five. Pick one. I'll give you a moment. The answer is two times three times five. The first has a one in it, and one is never a prime factor. The second stops early, because fifteen still splits.

Now the method, on eighty four. Start it wherever you like. Two and forty two. Four and twenty one. Six and fourteen. They all land on the same primes at the bottom. That is not luck. Every whole number is built from exactly one set of primes. That is what makes your answer checkable. Take six and fourteen. Six splits into two and three, both prime, so circle them. Then fourteen, and here is where it goes wrong. Fourteen gets written as seven and seven. It looks right, because seven plus seven is fourteen. But a tree multiplies. Seven times seven is forty nine. You will not catch that by staring at it. You catch it by multiplying the bottom row back. Two threes are six. Six sevens are forty two. Forty two sevens is two hundred and ninety four. Not eighty four. So fix that one branch. Fourteen is two times seven. The bottom row now reads two, three, two, seven. Two threes are six, six twos are twelve, twelve sevens are eighty four. That multiply-back is not an optional check. It is the last step of the tree. Do it every time, and a slip in the middle cannot survive into your answer. Write your primes smallest first. Two, two, three, seven. That is a correct answer. It is not yet an answer in index form. Index form is a tally. Count how many times each prime appears and write the count small and high. Two appears twice, so two squared. Three and seven appear once each. Eighty four equals two squared, times three, times seven. Same information, written the way the question asked for. Four words hold the method. Split, circle, check, tally. Split until everything is prime. Circle the primes. Check by multiplying back. Tally into index form.

Exam craft. Here is what examiners write, in their own words. Most students used a factor tree, but there were several instances of arithmetic errors. For example, ten was often split into five and five. Five and five is the mistake we just made with fourteen. Adding, when the tree multiplies. The check is what stops it. And on primes: having one correct prime factor was common as some candidates thought one was a prime number. Other errors came from giving factor pairs rather than two prime factors, so two and fifteen, or five and six, were seen. Both traps in one sentence. A one in the answer, or a factor pair left unsplit. You have seen both, on thirty. A third is reported with them: every prime found correctly, then the mark lost on the form of the answer. As a product of its prime factors means primes multiplied. In index form means use the powers.

Your turn. Sixty. Break it into primes, write it in index form, and multiply back to check you land on sixty. Off you go. Have a go at all three steps. I'll wait, and then we will work it through together. Sixty splits into six and ten. Six is two times three. Ten is two times five. Bottom row: two, three, two, five. Check it. Two threes are six, six twos are twelve, twelve fives are sixty. Now tally. Two appears twice, three once, five once. Sixty equals two squared, times three, times five.

So. Let's pull it together. Rational means you can write it as a fraction of two integers, recurring decimals included. Irrational means you cannot. Primes start at two. One is never prime. A factor pair is not a finished answer. And the method, in four words. Split, circle, check, tally.

That was video one of five. The next video takes two numbers, already broken into primes, and finds the biggest factor they share. It assumes you can build a tree. You can.

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