Class 6 Mathematics Chapter 5: Prime Time

Two lamps on a stage

Devika is looking after the lights for a school play. One green lamp blinks every 4 seconds. One red lamp blinks every 6 seconds. Both are switched on together at the count of zero. Devika wants to know when the stage will flash green and red at the very same instant.

The green lamp's counts

The green lamp blinks at 4, 8, 12, 16, 20 and 24 seconds, and onwards. These counts are the multiples of 4. A multiple of 4 is what you get when 4 is multiplied by a counting number. The list carries on for ever, because there is always a next counting number.

The red lamp's counts

The red lamp blinks at 6, 12, 18, 24, 30 and 36 seconds. These counts are the multiples of 6. Notice that the red marks are more spread out than the green ones. A bigger step size gives fewer blinks in the same stretch of time.

Numbers that stand in both lists

Read the two lists side by side. The numbers 12, 24, 36 and 48 stand in both of them. A number that is a multiple of 4 and also a multiple of 6 is called a common multiple of 4 and 6. At those counts the stage flashes green and red together.

The first time they agree

Of all the common multiples, 12 is the smallest. So the two lamps flash together for the first time at 12 seconds. After that they agree again every 12 seconds. That is why 24, 36 and 48 are on the list as well.

Common multiples never run out

Could Devika name the largest common multiple of 4 and 6? She could not. Add 12 to any common multiple and you have another one. The list 12, 24, 36, 48, 60 keeps growing for as long as you keep writing. Common multiples are endless, just as multiples are.

A tray of sweets

Now for a different sort of question. Meher has 18 laddoos and 30 pedas. She wants to fill identical plates. Every plate must hold the same number of laddoos, with none left over. Every plate must also hold the same number of pedas, with none left over. How many plates can she use?

Factors, in one line

A factor of a number divides it exactly, leaving no remainder. The factors of 18 are 1, 2, 3, 6, 9 and 18. The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. Each list stops, because a factor is never larger than the number it divides.

Numbers that divide both

Compare the two factor lists. The numbers 1, 2, 3 and 6 stand in both. These are the common factors of 18 and 30. So Meher can use 1, 2, 3 or 6 plates. With 6 plates, each one holds 3 laddoos and 5 pedas.

The quietest factor of all

Every factor list opens with 1, because 1 divides every number exactly. So any two numbers you choose already share a common factor, and that factor is 1. A pair that shares nothing beyond 1 has a name of its own, and the next topic gives it.

The largest common factor

The biggest number on the shared list is 6. It gives Meher the most plates she can fill. Unlike common multiples, common factors do come to an end, because nothing bigger than 18 can be a factor of 18.

Multiples march out, factors fit in

Keep the two ideas apart with one question. Am I multiplying or am I dividing? The multiples of 18 are 18, 36, 54 and onwards, marching away from 18. The factors of 18 are 1, 2, 3, 6, 9 and 18, all sitting at 18 or below.

Write the list, do not guess

Guessing a common factor is slow and it misses some. Write both factor lists in order, then ring what appears twice. For common multiples, write out a few multiples of the larger number, then test each one against the smaller. Nothing is left out that way.

A check before you finish

Take every answer back to the question. For a common factor, divide both numbers by it and look for a remainder of zero. For a common multiple, divide it by both numbers in the same way. If either division leaves something over, that number does not belong on your list.

Scenario:

Two bus routes leave the same stop. One goes every 12 minutes and the other every 18. They set off together at 6 am. A class is asked when they will next leave together. Imran says every 30 minutes, adding the two. Their teacher asks the class to list the departure times instead. The first route goes at 12, 24, 36, 48 and so on. The second goes at 18, 36, 54. They first meet again at 36 minutes. Imran sees that adding was the wrong operation. The answer is the lowest common multiple, because the buses meet where the two lists of multiples agree.

Analysis:

Adding the intervals gave 30, and the lists agree at 36. The meeting point is where both sequences land, which is what a common multiple is.

Activity:

Pick two intervals, such as 8 and 10 minutes. List the departure times of each and mark where they first meet.

Title:

Build the blinking strip

What You Notice:

The beans land on 12, 24, 36 and 48, spaced evenly at twelve boxes apart. The first bean marks the smallest common multiple, and the even spacing shows why adding 12 to one answer always gives another.

Stage crews really do plan flashing lights this way, choosing step sizes so that two lamps meet only rarely.

Bus timetables use the same idea: two routes leaving every 12 and every 20 minutes meet again only every 60 minutes.

Gear wheels with 8 and 9 teeth return to their starting position together only after 72 teeth have passed.

Telling multiples from factors — Factors fit inside, multiples march out — A factor sits at the number or below it, because it fits into the number. A multiple sits at the number or above it, because it is the number counted again.

Hunting for a common multiple — Start with the bigger step — List the multiples of the larger number first. There are fewer of them, so there is less work. Then test each one against the smaller number.

Rows on the school ground

The games teacher wants her class in equal rows, with no child left standing out. On Monday twenty children turn up. On Tuesday nineteen turn up. She finds that Monday is easy and Tuesday is oddly stubborn, and she cannot see why at first.

Twenty children, three shapes

With twenty she has a choice. One row of 20. Two rows of 10. Four rows of 5. Each choice is a pair of numbers whose product is 20. So each number used is a factor of 20, and the full list is 1, 2, 4, 5, 10 and 20.

Nineteen children, one shape

With nineteen she has no choice at all. One row of 19 is the only equal arrangement. Two rows would leave a child over, and so would three, four, five and every other size she tries. The factor list of 19 holds only 1 and 19.

What the rows were telling her

The number of shapes and the number of factors go together. A number with plenty of factors offers plenty of arrangements. A number with only 1 and itself offers exactly one. So counting factors settles the question, and no rows need drawing.

Exactly two factors

A number with exactly two different factors is called a prime number. Those two are always 1 and the number itself, with nothing between. So 19 is prime, and so are 2, 3, 5, 7, 11, 13, 17, 23 and 29. Count the factors and the answer must be two.

More than two factors

A number with more than two factors is called a composite number. It has at least one factor besides 1 and itself, and that extra factor is the second arrangement. So 20 is composite, and so are 4, 6, 8, 9, 10, 12 and 14.

Where 1 stands

The number 1 has one factor only, which is 1. That is fewer than two, so 1 is not prime. It is also not more than two, so 1 is not composite. The number 1 belongs to neither group, and it is not a small prime with a low count.

The only even prime

The number 2 has factors 1 and 2 only, so 2 is prime. Every other even number has 2 in its factor list as well as 1 and itself, which makes three factors at least. So no even number above 2 can be prime, and 2 stands alone.

Sifting the numbers to 50

Write 1 to 50 in a grid. Strike out 1, which is in neither group. Ring 2 and strike out every later multiple of 2. The next number still standing is 3, so ring it and strike out its later multiples. Do the same for 5, then for 7.

What is left standing

Fifteen numbers survive with a ring: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. Those are the primes up to 50. The 34 numbers struck out are the composites, and 1 sits apart from both.

Why the sieve works

A composite number has a factor besides 1 and itself, so it is a later multiple of some smaller number. That is exactly what gets struck out. A prime has no such smaller factor, so no strike ever reaches it and its ring survives.

Two numbers that share nothing

Now a question about pairs. Chandan has 8 red marbles and 9 green ones. He wants equal groups of each colour, using the same group size, and the size must be more than 1. He tries 2, 3 and 4 and every one of them fails.

Co-prime, stated carefully

The factors of 8 are 1, 2, 4 and 8. The factors of 9 are 1, 3 and 9. The only number on both lists is 1. Two numbers are co-prime when no number other than 1 divides both of them, so 8 and 9 are co-prime.

Co-prime does not mean both prime

Look again at 8 and 9. Neither one is prime, since each has more than two factors. They are still co-prime, because being co-prime is about what a pair shares and not about what either number is. Also 21 and 22 are co-prime, and 25 and 33.

Two different primes always are

Take 7 and 11. The factors of 7 are 1 and 7. The factors of 11 are 1 and 11. Their lists can meet only at 1, since 7 and 11 are not equal. So any two different primes are co-prime, while two composites may or may not be.

Scenario:

Sushila's grandmother runs a small hand-cranked flour mill. Inside it two toothed wheels turn against each other. The small wheel has 8 teeth and the large one has 9. Sushila paints one tooth on each wheel red, then lines the two red teeth up and begins to crank. She watches for the red teeth to meet again. They pass each other tooth after tooth, and only after 72 teeth have gone by do the two red marks come together once more. Her science teacher tells her this is deliberate. When the tooth counts share no factor above 1, every tooth of one wheel meets every tooth of the other in turn. That spreads the wear evenly, so no single tooth is ground down first.

Analysis:

A co-prime pair of tooth counts makes every tooth meet every other tooth, which spreads wear. The shared meeting point comes only at the product of the two counts.

Activity:

Cut two card discs, one with 8 notches and one with 9, mark one notch on each, and turn them together to count how many notches pass before the marks line up.

A sieve of this kind was worked out in Greece more than two thousand years ago and is still the quickest way to list small primes.

Primes come in pairs two apart surprisingly often: 11 and 13, 17 and 19, 29 and 31 are all such twins below 50.

Bank and internet security rests on the fact that multiplying two huge primes is easy while taking the product apart again is not.

Counting the factors — One, two, many — One factor means the number is 1, which stands alone. Exactly two factors means prime. Many, meaning more than two, means composite. Every whole number above zero lands in one of the three.

What co-prime is about — Co-prime is a pair word — A number is prime on its own, but co-prime always needs two numbers. Ask what the pair shares, not what either one is.

Farhan's chalk tree

Farhan is at the blackboard with the number 90. His teacher asks him to keep splitting it until he cannot split any more. He writes 90 at the top, draws two lines down from it, and writes 9 and 10 underneath. His classmate Neha would have started differently, and that turns out to matter less than either of them expects.

Start wherever you like

Farhan chose 9 and 10 because he spotted them first. He could have chosen 6 and 15, or 2 and 45. Any split into two factors is a fair start. The rule is only that the two numbers you write must multiply back to the number above them.

Split it or ring it

Now look at each new number. Is it composite? Then split it in turn. Is it prime? Then ring it and leave it alone. Farhan splits 9 into 3 and 3, and both of those are prime, so both get rings. He splits 10 into 2 and 5, and rings them too.

Reading the leaves

Four ringed numbers sit at the bottom of Farhan's tree: 3, 3, 2 and 5. Multiplying them gives 2 x 3 x 3 x 5 = 90, which is the number he started with. Writing a number as a product of primes like this is called its prime factorisation.

Neha takes another route

Neha starts 90 as 6 and 15 instead. She splits 6 into 2 and 3, and 15 into 3 and 5. Her leaves are 2, 3, 3 and 5. A third pupil starts 90 as 2 and 45, splits 45 into 5 and 9, then 9 into 3 and 3. The leaves are 2, 5, 3 and 3.

The same primes every time

Three different starts, and the same four primes at the end. Only the order on the page changed. This is the remarkable thing about prime factorisation: a number greater than 1 has exactly one, apart from the order in which you happen to write the primes.

Why the order is free

Multiplication may be carried out in any order without changing the answer. So 3 x 5 x 3 x 2 and 2 x 3 x 3 x 5 come to the same total. That is why three routes down the tree cannot disagree about what is at the bottom.

Writing it tidily

Since the order is ours to choose, we choose the tidiest: smallest prime first. So 90 is written 2 x 3 x 3 x 5, and 156 is written 2 x 2 x 3 x 13. Anyone comparing two factorisations then finds matching primes in the same place.

Prime factors, and the factorisation

These two phrases are not the same. The prime factorisation of 98 is 2 x 7 x 7. Its prime factors are just 2 and 7, because that list names the different primes once each. When a prime repeats, the factorisation is longer than the list of prime factors.

Where 1 and the primes stand

What line do you get for a prime such as 13? It is 13 itself, since there is nothing to break. And 1 has no prime factorisation at all, because no prime divides 1. That is one more reason 1 is kept out of the primes.

Building from two pieces

There is a shortcut when you already know the parts. Take 132 as 12 x 11. Now 12 is 2 x 2 x 3, and 11 is prime. Put the pieces together and 132 = 2 x 2 x 3 x 11. You need not go back to the top of the tree.

Multiply back, every time

Always finish by multiplying your primes together. For 120 the tree gives 2 x 2 x 2 x 3 x 5, and multiplying that returns 120. If the product is not the number you started with, a split went wrong somewhere and the line is worth checking.

Testing a pair for co-primes

Here is the first real use. Write 44 = 2 x 2 x 11 and 105 = 3 x 5 x 7. Not one prime turns up in both lines, so nothing above 1 can divide both numbers, and the pair is co-prime. Now try 52 = 2 x 2 x 13 and 91 = 7 x 13. The prime 13 appears in both, so this pair is not.

Testing whether one divides another

The second use is division. Is 210 a multiple of 15? Write 210 = 2 x 3 x 5 x 7 and 15 = 3 x 5. Both primes of 15 sit inside 210's line, and the primes left over are 2 and 7. So 210 = 15 x 14, and the answer is yes.

Counting the copies matters

One warning. Is 180 a multiple of 24? Write 180 = 2 x 2 x 3 x 3 x 5 and 24 = 2 x 2 x 2 x 3. Both lines use only 2s, 3s and a 5, so the primes look shared. But 24 needs three 2s and 180 has only two, so 24 does not divide 180.

Scenario:

A class wants to tile a rectangular board 24 centimetres by 36 with identical square tiles and no cutting. Nita tries 5 centimetre tiles and they do not fit either side. Their teacher asks the class which sizes could possibly work. A tile must divide both 24 and 36 exactly. The children list the common factors: 1, 2, 3, 4, 6 and 12. The largest is 12, which gives two tiles along one side and three along the other. Nita sees why 5 failed. It divides neither number, and a tile that does not divide a side will always leave a gap.

Analysis:

Five could not work for a reason, not by bad luck. A tile size has to divide both sides, which is what a common factor is.

Activity:

Choose a rectangle with whole-number sides. List the square tiles that would fit it exactly and name the largest.

Title:

A tree from a paper slip

Safety:

Take care with the scissors and cut away from your fingers.

What You Notice:

Both trees finish with three 2s, one 3 and one 5, however different the branches looked on the way down. Multiplying the ringed numbers together returns 120 each time, which is the check that the tree was drawn correctly.

Primes are treated as the building blocks of whole numbers, because every number above 1 is a product of them and of nothing else.

A prime factorisation is one of the few answers in mathematics that is exactly the same whichever route you take to reach it.

Splitting a very large number into its primes is so slow, even for computers, that online payments are guarded by that slowness.

Knowing when to stop breaking down — Split it or ring it — Look at each number on your tree. If it is composite, split it into two factors. If it is prime, ring it and leave it alone. You are finished when every branch end carries a ring.

Testing whether one number divides another — Count the copies — It is not enough for the same primes to turn up. Each prime must appear at least as many times in the larger factorisation as it does in the smaller one.

A long number on the board

Ismail's teacher writes 4,730 on the board and asks which of the numbers from 2 to 10 divide it. Ismail reaches for long division and starts on the first one. His teacher stops him. For several of these, she says, the answer is already on the board, if he knows where to look.

Dividing by 10, by eye

List the multiples of 10: 10, 20, 30, 40 and onwards. Every one of them ends in 0, and nothing else does. So a number is a multiple of 10 exactly when its last digit is 0. Ismail's number ends in 0, so 10 divides 4,730, giving 473.

Dividing by 5

Now list the multiples of 5: 5, 10, 15, 20, 25 and onwards. The last digits go 5, 0, 5, 0 for ever. So a number is a multiple of 5 exactly when it ends in 0 or in 5. That makes 4,730 and 4,735 both multiples of 5, while 4,736 is not.

Dividing by 2

The multiples of 2 run 2, 4, 6, 8, 10, 12 and onwards, and their last digits repeat 2, 4, 6, 8, 0. So a number is a multiple of 2 exactly when it ends in 0, 2, 4, 6 or 8. Here 4,730 passes and 4,735 fails.

Why one digit is enough for those three

Think about what a ten is worth. Ten is a multiple of 10, of 5 and of 2. So every ten, every hundred and every thousand inside a number is already divisible by all three. Only the units digit is left to spoil things, which is why reading it settles the question.

The last digit fails for 4

Try the same trick on 4 and it breaks. Look at 26 and 36. Both end in 6, but 36 is a multiple of 4 and 26 is not. Look at 14 and 24, or 18 and 28. In each pair the last digits agree and the answers differ, so one digit cannot be enough.

Two digits settle 4

The reason is that a single ten is not a multiple of 4, but a hundred is, since 100 = 4 x 25. So hundreds and thousands can never spoil a test by 4, and only the tens and units can. Read the number made by the last two digits and test that instead.

Testing 3,916 for 4

The last two digits of 3,916 make 16, and 16 is a multiple of 4. So 3,916 is a multiple of 4, and dividing gives 979. Now try 3,918. Its last two digits make 18, which leaves 2 over when divided by 4, so 3,918 is not a multiple of 4.

Three digits settle 8

For 8 the same argument shifts one place to the left. A hundred is not a multiple of 8, but a thousand is, since 1000 = 8 x 125. So thousands and everything above them are safe, and the last three digits carry the whole test.

Testing 7,432 for 8

The last three digits of 7,432 make 432. Dividing 432 by 8 gives 54 exactly, so 7,432 is a multiple of 8, and the full division gives 929. Compare 7,428. Its last three digits make 428, which leaves 4 over, so 8 does not divide it.

Passing one test and failing another

These tests are linked, but not equal. Any multiple of 8 is also a multiple of 4 and of 2, because 8 already holds them. The other way round can fail. The number 3,916 passes the test by 4, yet 916 leaves 4 over when divided by 8, so it fails there.

What is not here

Only 10, 5, 2, 4 and 8 have been tested this way. There are short tests for 3, 6, 7 and 9 as well, and the book keeps those for a later class. Until then, those factors are found by dividing, and no guess about their last digits will do.

Which number looks special?

Now a puzzle to finish with. Four numbers sit in a box: 7, 18, 25 and 32. Which one looks special to you? Different answers are all correct here, because each of the four has something that none of the others has.

Four numbers, four reasons

Amrita picks 7, the only prime and the only single-digit number. Joseph picks 18, the only multiple of 3. Lata picks 25, the only multiple of 5 and the only square. Faiz picks 32, the only multiple of 4. Nobody is wrong, and comparing reasons teaches more than any single answer.

A grid of primes

Here is the last puzzle. Fill a two by two grid with primes so that each row multiplies to the number beside it and each column to the number beneath it. The rows must give 10 and 21. The left column must give 6 and the right column 35. The answer is 2 and 5 on top, with 3 and 7 below.

Scenario:

A women's co-operative in a small town makes amla candies and supplies them to a school kitchen. This month they have made 5,624 candies. The kitchen asks for them in packets of 8, with no candy left loose, because each child is to receive one packet. The youngest member of the co-operative starts a long division on a slate. Her aunt stops her and points at the last three digits instead. Dividing 624 by 8 gives 78 exactly, so the whole batch will pack with nothing over. The aunt explains that a thousand candies always split into eights, so the thousands in the count can be ignored altogether and only 624 needs testing.

Analysis:

A divisibility test turns a long division into a glance, because a thousand is a multiple of 8 and so cannot affect the answer. The test is a real time-saver in everyday work.

Activity:

Count the pencils or beans in a jar, write the total down, and use the tests for 2, 4, 5, 8 and 10 to list which group sizes would share them out with nothing left over.

Shopkeepers and clerks used these tests long before calculators, to check a bill's total without dividing it out.

Every test here reads the digits from the right, which is why the huge left-hand part of a number can simply be ignored.

The tests for 3, 6, 7 and 9 exist too, and the book saves them for a later class rather than leaving them out of mathematics.

How many digits each test reads — One, two, three: ten, four, eight — One digit for 10, 5 and 2. Two digits for 4. Three digits for 8. The harder the number is to divide by, the further left you have to look.

Passing 8 and passing 4 — Eight carries four with it — Any multiple of 8 is also a multiple of 4 and of 2, because 8 itself holds them. A multiple of 4 need not be a multiple of 8, though.