Factorise and regroup instead of multiplying head-on

0:00Here is a multiplication. One hundred and sixteen, times five. You could set it out and grind through it. Six fives are thirty, carry the three, and so on. Or you could do it in one line, in your head, right now. Halve one hundred and sixteen. That is fifty eight. Now put a zero on the end. Five hundred and eighty. That is the answer. Exactly the answer, not close to it.

0:29And it cost one halving and one shift. Most people meet this as a trick for five, and file it away as a trick for five. It is not. It is a fact about ten, and by the end of this you will know exactly which numbers have one and which never will. So why does it work? Because five is not really the number doing the work here. Ten is.

0:55Ten halved is five. Those are two names for the same number. So multiplying by five, and multiplying by ten halved, are the same instruction written two ways. And once it is written the second way, you get to choose which part to do first. Halve, then shift. One hundred and sixteen halves to fifty eight, and fifty eight times ten is five hundred and eighty. Nothing was rounded and nothing was thrown away.

1:27That is worth saying, because dividing by two feels like it ought to make the answer smaller. It does. And then multiplying by ten makes it bigger again, by more. The two moves are one move, written in two steps. Now watch the same idea one size up. Eight hundred and twenty four, times twenty five. That is not a multiplication anybody wants to do in their head. But twenty five is a hundred quartered.

1:59So quarter eight hundred and twenty four. That gives two hundred and six. And then shift by two places. Twenty thousand six hundred. The same three moves. Rewrite the awkward factor, divide, shift. And notice the shift grew with the factor. Five wanted one zero. Twenty five wants two. That is the first hint that this is not a collection of separate tricks. It is one idea at different scales.

2:31Let's say plainly what is free here. Multiplying by ten, or a hundred, or a thousand, costs you nothing at all. And it is worth being careful about why. People say that multiplying by ten adds a zero. It does not. It moves every digit one place to the left. The five in fifty eight was worth fifty. After the shift, that same five is worth five hundred. The zero was not added by us. It is what fills the units place once everything has moved out of it.

3:08That sounds like a fussy distinction, and it is not. The moment a number has a decimal point, adding a zero stops working, and shifting still works. So shifts are free. Everything in this video is about turning a hard multiplication into a shift. There is a second freedom here, and it is the one people forget they have. When you multiply several numbers together, you do not have to take them in the order somebody wrote them.

3:41Here is two, times one thousand seven hundred and sixty eight, times fifty. Read it left to right and your first job is doubling one thousand seven hundred and sixty eight. Then you still have to multiply that by fifty. Instead, look straight past the middle number. Two times fifty is a hundred. So the whole thing is one thousand seven hundred and sixty eight, times a hundred. One lakh seventy six thousand eight hundred.

4:12You never did a multiplication. You spotted a pair, and then you shifted. The order was never a rule. It was just the order somebody happened to write it in. So let's collect the factors that behave like this. Five is ten over two. Twenty five is a hundred over four. A hundred and twenty five is a thousand over eight. There is the pattern, and it is a pleasing one. Each time the power of ten goes up a step, the divisor doubles.

4:46But those three are not the only ones. Fifty is a hundred over two. Two hundred and fifty is a thousand over four. Two thousand five hundred is ten thousand over four. Every row has the same shape. A power of ten, divided by something small. And that is what a convenient factor means in this whole topic. Not a number that is easy by itself, but a number that is a power of ten in disguise.

5:18Which raises the obvious question, and we will get to it. Which numbers are? First, watch what happens when you use both freedoms at once. A hundred and twenty five, times forty, times eight, times twenty five. Four numbers. Head on, that is three multiplications, and not one of them is pleasant. So do not go head on. Look for pairs. A hundred and twenty five times eight is a thousand.

5:49And forty times twenty five is also a thousand. So the whole thing is a thousand times a thousand. Ten lakh. Three ugly multiplications became one easy one, and the only thing that made it possible was refusing to work left to right. That is the real skill in this topic. Before you calculate anything at all, look at what you have been handed. Now the question that has been sitting there all along. Does every number have a shortcut like this?

6:24Let's try three. For three to work, three would have to divide into some power of ten exactly. Ten divided by three. Three, remainder one. A hundred divided by three. Thirty three, remainder one. A thousand divided by three. Three hundred and thirty three, remainder one. It is always remainder one. And it always will be, because ten itself leaves remainder one, and every power of ten is just ten multiplied by itself again.

6:59So there is no rewriting for three. Not a clumsy one, not a big one. None, at any size. And exactly the same thing happens for seven, and for eleven, and for thirteen. So which numbers do work? Look at what a power of ten actually is, underneath. A hundred is ten times ten. And ten is two times five. So a hundred is two twos and two fives, and nothing else at all. A thousand is three of each.

7:33Which means anything that divides a power of ten can only be built out of twos and fives. And that is the entire answer. One, two, four, five, eight, ten, sixteen, twenty, twenty five, thirty two, forty, fifty, sixty four, eighty, a hundred, a hundred and twenty five. Every one of those has a shortcut waiting for it. Anything carrying a three, or a seven, does not, and never will.

8:03Six looks friendly and is hopeless, because six is two times three, and that single three ruins it. One honest thing before we finish. Every example so far divided perfectly, and that was not luck. They were chosen. So try eight hundred and twenty three, times twenty five. Twenty five is still a hundred over four, so the method has not changed at all. Eight hundred and twenty three, shifted two places, is eighty two thousand three hundred.

8:38Now divide that by four. Twenty thousand five hundred and seventy five. The shortcut did not fail. It handed you the exact answer, in one division. What it stopped being was mental. And that is a completely different complaint from being wrong. Last thing. Run the whole idea backwards. Somebody hands you an answer, twelve crore, and asks for two numbers that multiply to make it. Twelve crore is twelve, followed by seven zeros.

9:12So twelve times one crore works. But so does one thousand two hundred, times one lakh. And so do dozens of others, because you are free to choose how many of those zeros go to each side. That is the same skill as before, just pointed the other way. You are deciding where to put the powers of ten. Here is one to try. Take any multiplication with a five, a twenty five or a hundred and twenty five in it, and get it down to one line.

9:44Then pick a number that has no shortcut at all, and convince yourself that it never will have one.