Class 7 Mathematics Chapter 11: Finding Common Ground

Two lengths, one shared measure

A carpenter has two planks, 24 cm and 36 cm long. She wants to cut both into equal pieces, with no wood wasted and every piece the same length.

How long can each piece be? The answer has to divide both 24 and 36 exactly.

Factors

A factor of a number divides it exactly, leaving nothing over. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Common factors

Now compare the two lists. The numbers appearing in both are 1, 2, 3, 4, 6 and 12.

Each of those is a common factor, so each is a piece length that works for both planks.

The highest of them

The carpenter wants the longest pieces, so she wants the largest of those common factors, which is 12.

That number is called the highest common factor of 24 and 36, written HCF for short.

Listing gets slow

For small numbers, listing every factor works well enough. For large ones it is slow, and it is easy to miss one.

So there is a better method, and it uses prime numbers.

Prime factorisation

Every number can be broken down into prime factors in exactly one way. Take 18, which is 2 times 9, and 9 is 3 times 3.

So 18 is one 2 and two 3s. In the same way 45 is 3 times 15, and 15 is 3 times 5, so 45 is two 3s and one 5.

Taking what both have

Now line the two breakdowns up. The number 18 has one 2 and two 3s. The number 45 has two 3s and one 5.

The 2 appears only in 18, so it cannot help. The 5 appears only in 45. The two 3s appear in both.

So the HCF is built from the shared part

Multiply the primes that both numbers have, as many times as both have them. Here that is 3 times 3, which is 9.

So the HCF of 18 and 45 is 9. No listing was needed at all.

Take the smaller count of each prime

Notice the rule carefully. If one number has three 2s and the other has one, the HCF takes just one 2.

It takes the smaller count every time, because the HCF must divide both numbers.

Checking the method on the planks

Try it on 24 and 36 to see the methods agree. The number 24 is three 2s and one 3, and 36 is two 2s and two 3s.

Both have at least two 2s and at least one 3, so the HCF is 2 times 2 times 3, which is 12. That matches the listing.

When nothing is shared

Now take 8 and 15. The number 8 is three 2s, and 15 is one 3 and one 5.

There is no prime in both breakdowns, so there is nothing to multiply. The HCF is 1.

Coprime numbers

Two numbers whose HCF is 1 are called coprime. They share no factor except 1.

This does not mean either is prime. Both 8 and 15 have factors of their own, but none in common.

What an HCF is for

The planks were one use: the largest equal piece that divides several quantities. The same question appears in tiling, packing and sharing.

It is also what reduces a fraction to its lowest terms, by dividing top and bottom by their HCF.

Two checks worth making

First, the HCF must divide both numbers exactly. Twelve divides 24 twice and 36 three times, so that passes.

Second, nothing larger must do so. The next common factor down from 12 was 6, and no common factor above 12 appeared in either list.

Why it can never exceed the smaller number

Here is a useful limit. The HCF has to divide the smaller number, so it can never be bigger than it.

For 18 and 45 the HCF is at most 18, and it turned out to be 9. If it ever equals the smaller number, that number divides the larger one.

What carries forward

A common factor divides both numbers, and the highest of them is the HCF. Listing the factors works for small numbers, and prime factorisation works for any: take each shared prime the smaller number of times.

Two numbers with HCF 1 are coprime, which does not make either of them prime. An HCF can never exceed the smaller of the two numbers.

Scenario:

A carpenter has planks of 24 cm and 36 cm and wants to cut both into equal pieces with nothing wasted and the pieces as long as possible. An apprentice lists the factors of each and answers 6. For a second job, with lengths of 18 cm and 45 cm, he multiplies every prime he can see and answers 90. For a third job, with 8 cm and 15 cm, he says no cutting is possible because the two share nothing. The carpenter says two quick checks would have caught all three.

Analysis:

One answer is a common factor but not the highest, one exceeds both numbers, and one misreads an HCF of 1.

Activity:

Measure five pairs of objects, find the HCF of each pair by primes, and check every answer divides both and is the largest that does.

Title:

Cut and compare

Safety:

Take the usual care with the scissors. No other hazards.

What You Notice:

The ringed lengths are 1, 2, 3, 4, 6 and 12, and cutting at 12 leaves nothing over on either strip. The primes give 9 for the second pair, and the third pair shares no prime at all, so its HCF is 1.

Coprime says nothing about either number being prime: 8 and 15 are coprime and both have factors of their own.

Dividing the top and bottom of a fraction by their HCF is exactly what reduces it to its lowest terms.

An HCF can never exceed the smaller of its two numbers, which rules out a wrong answer at a glance.

Shared primes only, at the smaller count.

An HCF never beats the smaller number.

Two bells ringing together

A school has two bells. One rings every 6 minutes and the other every 8 minutes. They ring together at the start of the day.

When will they next ring together? The answer must be a whole number of 6 minute gaps and also a whole number of 8 minute gaps.

Multiples

A multiple of a number is what you get by multiplying it by a whole number. The multiples of 6 are 6, 12, 18, 24, 30, 36 and so on.

The multiples of 8 are 8, 16, 24, 32, 40, 48 and so on.

Common multiples

Now look for numbers in both lists. The first is 24, and 48 comes next.

Each of those is a common multiple, and each is a time when both bells ring.

The least of them

The question asked when they would NEXT ring together, so we want the smallest of the common multiples, which is 24.

That is the lowest common multiple of 6 and 8, written LCM for short. The bells ring together every 24 minutes.

There is no greatest common multiple

Notice something about the list of common multiples. After 24 comes 48, then 72, and it never stops.

So there is no highest common multiple to ask for, which is why only the lowest one is given a name.

Listing works, but not for long

As with factors, listing multiples is fine for small numbers. For large ones the lists get long before they meet.

So the prime method is better here too.

Prime factorisation again

Take 12 and 18. The number 12 is two 2s and one 3, and 18 is one 2 and two 3s.

A common multiple has to be divisible by both, so it must contain enough of every prime to cover each number.

Taking the larger count this time

For the 2s, the number 12 needs two of them and 18 needs one, so two 2s will cover both. For the 3s, 18 needs two and 12 needs one, so two 3s are needed.

Multiplying gives 2 times 2 times 3 times 3, which is 36. So the LCM of 12 and 18 is 36.

Why the larger count

The reason is the opposite of the HCF rule, and it is worth saying plainly. An HCF has to DIVIDE both numbers, so it can never take more of a prime than a number has.

An LCM has to be DIVISIBLE BY both numbers, so it must take at least as many as the greedier one needs.

And lone primes are kept

For the same reason, a prime held by only one of the numbers must still appear. Leaving it out would make the answer fail to be divisible by that number.

So an LCM keeps everything, at the larger count, while an HCF keeps only what is shared, at the smaller count.

When the numbers are coprime

Take 5 and 7, which are both prime and share nothing. The LCM must contain one 5 and one 7.

So it is 35, which is simply 5 times 7. Whenever two numbers are coprime, their LCM is their product.

Why that happens

With nothing shared, no prime is covered twice over. Every prime in either number has to appear, and none of them overlaps.

Multiplying the two numbers is then exactly what the rule asks for.

What an LCM is for

The bells were one use: when two repeating events coincide. Gears, timetables and shift patterns all raise the same question.

It is also the common bottom number used when adding fractions, and the lowest one is the LCM of the two bottoms.

Checks on an LCM

Two checks are worth making. First, both numbers must divide the answer exactly: 6 and 8 both divide 24.

Second, nothing smaller must. Twelve is divisible by 6 but not by 8, so 24 is indeed the least.

A limit worth knowing

An LCM can never be smaller than the larger of the two numbers, because it has to be a multiple of it.

It also can never exceed the product of the two. For 6 and 8 that puts the answer between 8 and 48, and 24 sits inside.

What carries forward

A common multiple is divisible by both numbers, and the least of them is the LCM. There is no greatest, which is why only the least has a name.

By primes, take every prime at the LARGER count and keep the lone ones, which is the exact opposite of the HCF rule. Coprime numbers have their product as their LCM, and any LCM lies between the larger number and the product.

Scenario:

A school has bells ringing every 6 and every 8 minutes, starting together, and the caretaker needs to know when they next coincide. A helper lists multiples and answers 48. For a pair of gearwheels with 12 and 18 teeth, the helper breaks both into primes but takes the smaller count of each, answering 6. For two cogs with 5 and 7 teeth he says no answer exists, since they share nothing. He also asks the caretaker for the greatest common multiple of 6 and 8.

Analysis:

One answer is a common multiple but not the least, one uses the HCF rule by mistake, one misreads a coprime pair, and one question has no answer.

Activity:

Time two repeating events at home, work out when they coincide by both methods, and check that both numbers divide your answer.

Title:

Ring the changes

Safety:

Nothing hazardous here.

What You Notice:

The doubly marked points are 24 and 48, so the LCM of 6 and 8 is 24. The primes give 36 for 12 and 18, and both numbers divide it. For 5 and 7 the answer is 35, which is exactly their product, because they share no prime.

There is no greatest common multiple of two numbers, which is why only the lowest one has a name.

The LCM and HCF rules are exact opposites: larger count against smaller, and keeping lone primes against dropping them.

The lowest common bottom number for adding two fractions is the LCM of the two bottoms.

An LCM keeps everything, at the larger count.

HCF divides into, LCM is divided by.

A pattern worth noticing

Take 20 and 30. Their HCF is 10 and their LCM is 60.

Now multiply those two answers together. Ten times 60 is 600, and 20 times 30 is 600 as well.

Checking it on a second pair

That could be luck, so try another pair. Take 14 and 21, whose HCF is 7 and whose LCM is 42.

Seven times 42 is 294, and 14 times 21 is 294. The same thing has happened again.

The relation

For any two numbers, the HCF multiplied by the LCM equals the two numbers multiplied.

It holds every time, and the reason comes straight from the two rules of the last two topics.

Why it works out

Take any prime, and count how many times it appears in each number. The HCF takes the SMALLER of those two counts and the LCM takes the LARGER.

A smaller count plus a larger count is just the two counts added together.

Every prime is counted the right number of times

So when the HCF and LCM are multiplied, each prime appears as many times as it did in the two numbers altogether.

That is exactly what multiplying the two numbers gives. The relation is not a coincidence at all.

Using it as a shortcut

The relation saves work. Find the easier of the two answers, then get the other by arithmetic.

For 20 and 30, their product is 600. If the HCF of 10 is found first, the LCM is 600 divided by 10, which is 60.

Which one to find first

An HCF is usually the easier one. It is smaller, so a small mistake shows up quickly, and the prime method needs only the shared part.

So find the HCF, then divide the product by it to get the LCM.

A check on a pair of answers

The relation is also a check. If you have found both an HCF and an LCM, their product must match the product of the numbers.

For 14 and 21 we had 7 and 42, and 7 times 42 matched 294. Had it not, one of the two answers would be wrong.

When one number divides the other

Try 7 and 21. Since 7 divides 21, the HCF is 7 and the LCM is 21.

The relation holds: 7 times 21 is 147, and 7 times 21 is the product too. The HCF is the smaller number and the LCM is the larger.

Coprime pairs again

For a coprime pair the HCF is 1, so the relation says the LCM is the product divided by 1.

That is the product itself, which is exactly the rule found in the last topic. The relation explains it.

A quicker procedure for an HCF

There is also a way to find an HCF with no prime factorisation at all. Take 20 and 30, and subtract the smaller from the larger.

That leaves 10, so now consider 20 and 10. Subtract again, leaving 10, so now consider 10 and 10.

Stopping when they match

Once the two numbers match, that number is the HCF. Here both are 10, and the HCF of 20 and 30 is indeed 10.

Try it on 14 and 21. Subtracting gives 7, then 14 and 7 gives 7, so both are 7 and the HCF is 7.

Why subtracting keeps the HCF

The procedure looks like a trick, so it is worth seeing why it works. Suppose some number divides both 20 and 30.

Then it divides their difference of 10 as well, because taking away whole lots of it leaves whole lots of it.

And nothing is gained either

The reverse holds too. Any number dividing both 10 and 20 also divides 30, since 30 is 10 plus 20.

So the two pairs have exactly the same common factors, and therefore the same highest one. Every step preserves the answer.

Which method to use when

Prime factorisation shows the structure and gives the LCM as well. Subtracting is faster when the numbers are awkward to break down.

The relation then supplies the LCM from the HCF, so either route reaches both answers.

What carries forward

The HCF multiplied by the LCM equals the two numbers multiplied, because each prime is taken at the smaller count and the larger, which add to the two counts.

That gives a shortcut, since the easier answer yields the other, and a check on any pair of answers. An HCF can also be found by repeatedly subtracting the smaller from the larger until the two match, which works because every step keeps the common factors unchanged.

Scenario:

A pupil finds the HCF of 20 and 30 to be 10 and the LCM to be 60, and notices that 10 times 60 matches 20 times 30. She asks whether that always happens. A classmate says it is a coincidence of round numbers. She then has to find the LCM of 14 and 21, whose HCF she has already worked out as 7. Another classmate reports the HCF of 20 and 30 as 1, having run the subtracting procedure all the way down past 10.

Analysis:

One question has a reason behind it, one claim is wrong, one answer can be had by division, and one procedure was stopped too late.

Activity:

Take five pairs of numbers, find each HCF by subtracting and each LCM by the relation, then check every pair of answers against the product.

Title:

Two routes to both answers

Safety:

Nothing hazardous here.

What You Notice:

Both routes give an HCF of 10 for the first pair and 7 for the second. Dividing the products by those gives 60 and 42, matching the prime method. Every HCF times LCM comes back to the product of its two numbers.

The relation explains the coprime rule: an HCF of 1 makes the LCM the product itself.

The subtracting procedure finds an HCF without breaking either number into primes at all.

Every step of that procedure leaves the whole set of common factors unchanged, which is why the answer survives.

HCF times LCM is the product.

Stop when the two numbers match.