PrepShorts · Study sheet · Class 7 Mathematics · Chapter 3, Finding Common Ground
Chapter 3 · Finding Common Ground
Getting the HCF and the LCM out of one division ladder
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A layout arrives with nothing on it labelled: 84 and 180 at the top, some numbers down the left, a closing row underneath.
The idea
The stacked-division layout is not a third method to memorise alongside the two already learnt — it is those two methods being run at the same time on the same sheet. Peeling a common factor off both numbers at every step sorts the primes into two piles: the ones both numbers own end up in the left column, and whatever is left unshared ends up in the bottom row. So the left column multiplies to the HCF and the whole L-shape — column plus bottom row — multiplies to the LCM, from one calculation. Seeing that is also what makes Guna's shortcut legitimate rather than a trick: nothing in the argument ever required the divisor to be prime, only that it divide both.
What you should be able to do
- Read a stacked-division layout and say what each row and each divisor records
- State the condition for taking another step, and the condition for stopping
- Carry out the layout for a pair of two- or three-digit numbers
- Read the HCF off the finished layout, and explain why the left column gives it
- Read the LCM off the same finished layout, and explain why the bottom row has to be included
- Explain why dividing by a larger common factor in one step is allowed
- Predict what happens to the two answers if the divisors are chosen in a different order, and check the prediction
- Choose between the layout and the two separate factorisations for a given pair
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| division method | the layout that records each divisor at the left and each quotient below | printed in bold in §3.1, Part II, p.50, and extended to pairs in §3.3, p.61 |
| quotient | the result of one division, written on the next row | printed in §3.3, Part II, p.61 |
| common prime factor | a prime that divides both numbers, and so can be taken out of the pair | printed in §3.3, Part II, p.61 |
| common factor | a number that divides both of the given numbers, prime or not | printed throughout the chapter, and again in §3.3, Part II, p.62 |
| Highest Common Factor (HCF) | the largest of the common factors | printed in bold in §3.1, Part II, p.48 |
| Lowest Common Multiple (LCM) | the smallest of the common multiples | printed in bold in §3.2, Part II, p.56 |
| procedure | the chapter's word for a laid-out method you can follow step by step | printed in the subheading "Efficient Procedures for HCF and LCM", §3.3, Part II, p.60 |
| division ladder | the explanation's name for the stacked layout used on a pair of numbers | an added label; the chapter prints "division method" for the single-number version and gives the paired version no name of its own (all twenty pages, pp.47–66, read) |
| the L | the explanation's shorthand for the left column plus the bottom row, whose product is the LCM | an added shorthand, not printed in this chapter — but the book does draw exactly this shape, in blue, around the LCM layouts on p.61 (verified on p.61) |
Where people slip up
- "The layout is a new rule you have to learn." It is bookkeeping for the two rules already established. Every divisor written at the left is a factor both numbers still share; the column is therefore the shared part and nothing else. Show the two separate factorisations beside the layout once, and the point is made.
- "The divisor has to be prime." The chapter's own two characters break that and the chapter endorses them: any common factor may be taken out, and 50 and 10 are not prime. What has to be true is only that the divisor divides both numbers. Prime divisors are the safe default because you cannot overshoot with them, not because larger ones are wrong.
- "Different choices of divisor give different answers." They give different ladders and the same two answers. This is the same stability the chapter met in Breaking a number down to primes, and why the result is unique, and section 10's interactive figure exists to make it visible.
- "The LCM is the product of the divisors, like the HCF." Then 300 and 150 would give 150 for both. The closing row is carrying every prime that was never shared, and dropping it drops exactly those.
- "You can stop as soon as one of the two numbers becomes prime." You stop when the pair has no common factor left. In the 300 and 150 layout the closing row is 2, 1 — a division by 3 was still available at 6, 3 even though 3 is prime.
- "The layout is always faster." For a pair with no shared factors at all it stops immediately and tells you almost nothing beyond the HCF being 1; the LCM then still needs the product. Say when to reach for it.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 6 Q3
Transcript1,347 words
Here is a layout, and nothing on it is labelled. Eighty-four and one hundred and eighty at the top. A two down the left. Forty-two and ninety underneath. Another two. Twenty-one and forty-five. A three. Seven and fifteen. And then it stops. And no answer is written anywhere on it. Not at the bottom, not at the side, nowhere. So the question is not how to copy it. It is what it is doing, and why it stops exactly there.
Because this is not a third method to learn alongside the two you already have. It is both of those methods, running at the same time, on one sheet. Every step does one thing. Find something that divides both numbers, and divide both by it. Eighty-four and one hundred and eighty are both even, so take out a two. Forty-two and ninety. Both still even. Take out another two. Twenty-one and forty-five.
Both divide by three. Seven and fifteen. Now, seven and fifteen. Is there any number bigger than one that divides both of those? No. So there is nothing left to take, and that is the stopping condition. Not when a row gets small, and not when a row turns prime. You stop when the two numbers have nothing left in common. Look at what the left-hand column has collected. Two, two, three.
Now factorise the two numbers separately, the slow way. Eighty-four is two, two, three, seven. One hundred and eighty is two, two, three, three, five. Two twos in both. One three in both. And nothing else in both. Two, two, three. That is not nearly the same list. It is the identical list. Every step took out something both numbers were holding, and it kept going until neither held anything more.
So the column cannot be anything except the shared part. Which means the column is the highest common factor, already built. Two times two times three. Twelve. And notice what you did not have to do to get it. Nothing was factorised, and nothing was compared. Compare that with the work you would otherwise do. Break both numbers into primes, line them up, and take each shared prime at its smaller count.
The ladder did exactly that. It just did it one row at a time, and wrote the answer down the side. So one column, one multiplication, and the first of the two answers is already out. Twelve is the highest common factor of eighty-four and one hundred and eighty. Now the second answer, out of the same picture. Two more ladders, so there is something to read them against. Three hundred and one hundred and fifty. Take out a two: one hundred and fifty, seventy-five.
A five: thirty, fifteen. Another five: six, three. Six and three still share a three, even though three is prime. Take it out. Two and one. Two and one share nothing, so it stops. Six hundred and thirty, and seven hundred and seventy. A two: three hundred and fifteen, three hundred and eighty-five. A five: sixty-three, seventy-seven. A seven: nine and eleven. Nine and eleven share nothing. Stop. Both columns give their highest common factors. Two, five, five, three is one hundred and fifty. Two, five, seven is seventy.
But there is more written on each of those sheets than just the column. There is a bottom row, sitting there, that we have not used for anything. Two and one, at the foot of the first. Nine and eleven, at the foot of the second. So take the column, turn the corner at the foot, and pick up the bottom row as well. That shape is an L. Multiply everything inside it.
One hundred and fifty, times two, times one. Three hundred. Seventy, times nine, times eleven. Six thousand nine hundred and thirty. Those two are the lowest common multiples. And the bottom row is the reason. Think about what the ladder did to each individual prime inside the two numbers. If both numbers held it, some step took it, and it is in the column. If only one of them held it, no step could ever take it, so it is still sitting in the bottom row at the end.
Between them, the column and the bottom row hold every prime that was ever there. The column has the shared ones. The bottom row has everything that was never shared. And that is the recipe for the lowest common multiple: every prime, at its larger count. Drop the bottom row and you get the highest common factor back instead. For six hundred and thirty and seven hundred and seventy, that answer is ninety-nine times too small.
Now for something that looks, at first, like cheating. Three hundred and one hundred and fifty took four steps. Two, five, five, three. But fifty divides both of them. So take out fifty, in one go. Six and three. Then a three. Two and one. Two steps instead of four, and the same bottom row. The column is now fifty times three, which is one hundred and fifty. The same first answer.
And the L is one hundred and fifty, times two, times one. Three hundred. The same second answer. Fifty is not prime. It is two, five, five — the first three rungs of the long ladder rolled into one. The same move on the other pair. Six hundred and thirty and seven hundred and seventy both end in a zero, so ten divides both. Sixty-three and seventy-seven. Both divide by seven. Nine and eleven.
Two steps, and the same foot as the three-step version. The column is ten times seven. Seventy. The L is seventy, times nine, times eleven. Six thousand nine hundred and thirty. Both answers identical. Neither ten nor fifty is prime, and neither of them caused the slightest trouble. So what does one step of this thing actually require? One thing only. Whatever you take out has to divide both numbers.
Nothing anywhere in the argument asked for a prime. It asked for something shared, and it kept asking until nothing was shared. And there is a reason a bigger step is safe. Pull a factor out of both numbers, and the highest common factor comes out with it, divided by exactly that factor. Take fifty out, and what is left has a highest common factor fifty times smaller. Nothing is lost; it moved into the column.
So a big step is not skipping work. It is several rungs done at once. The only thing primes buy you is that you can never take out more than is actually there. Which raises a worry. If you get to choose, do different choices give different answers? Three hundred and one hundred and fifty can be laddered forty-four different ways. Four rungs, taking primes. Three rungs. Two rungs. Or one single rung of one hundred and fifty, taken all at once.
Every single one of them ends at two and one. Every column multiplies to one hundred and fifty. Every L multiplies to three hundred. The route is yours. The foot is not. And it has to be that way, because the two answers were never about the ladder. They were about the two numbers. The ladder is only bookkeeping. One last question, which is when you should actually reach for this.
When the two numbers share a lot, it is the best tool there is. Both answers, one calculation, nothing factorised. When they share nothing, it is the worst. Take a pair with no common factor and the ladder stops before it starts. No rungs at all. Highest common factor one, and it has told you nothing you could not already see. That is not a rare case either. Below one hundred and twenty, more than half of all pairs are like that.
Two to try. Ninety and one hundred and fifty. Eighty-four and one hundred and thirty-two. Run the ladder on each, and read both answers straight off the one picture. Thirty and four hundred and fifty. Twelve and nine hundred and twenty-four.
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.
Builds on
- Breaking a number down to primes, and why the result is uniqueClass 7 · Ch 3, Finding Common Ground
- HCF: take the fewest occurrences of each primeClass 7 · Ch 3, Finding Common Ground
- LCM: take the most occurrences of each primeClass 7 · Ch 3, Finding Common Ground
Either side of this one
- What HCF and LCM do for consecutive, even, and co-prime numbersClass 7 · Ch 3, Finding Common Ground
- Conjecture and generalisation: what mathematicians mean by those wordsClass 7 · Ch 3, Finding Common Ground