Prime Building Blocks

Key Stage 3

Maths

Prime factor decomposition

Which of these could not be a value of b?

﻿Hello, I'm Mr. Langton, and today we're going to be looking at prime building blocks. All you're going to need is something to write with and something to write on. Try and find a quiet space where you won't be disturbed. And when you're ready, we'll begin. We'll start with the try this activity. On the grid, I have highlighted every number that can be made, as a product of ones and twos. For example, one times two, two times two, two times two times two, two times two times two times two times two. There's one more as well, two times two times two times two times two so you get 32. I was wondering what other numbers you can make, if I let you multiply ones, twos and threes. Pause the video and have a go. When you're ready, un-pause it and we can look at it together. You can pause in 3, 2, 1. Okay, some examples of numbers you could make. We could do one times three, to get three. We could do two times three, to get six. We could do three times three, to get nine. We could do three times three times three, to get 27. We could do two times two times three, which would be 12. We could do two times three times three, which is 18. We could do two times two times three times three, that will be 36. And there's probably a few others that you're not able to think of as well. Can you get those? Can you get any more? Let's have a look at what we could do next. So here's the grid with all the products of ones, twos and threes. Yasmine says, I don't think this list will change if we include fours as well. So have a think about that. What about one times four? That would be four and that's already on the grid. What about two times four? That'll be eight. That's already on the grid. What about two times two times four? That'd be 16, which is already on the grid. Yasmine seems to be right. Maybe we could try three times four, couldn't we? That's 12. Three times three times four. That would be 36, that's already on the grid. So Yasmine seems to be right. Can you see why she's right? The answer is because the number four is made up of two times two. Since we've already been using twos as we go along, that means that any answer that would involve multiplying by four could already be made by doing two multiplied by two. And the reason for that is that four is not a prime number. Four is not prime. Two is prime, and three is prime. So we can use those as building blocks to make other numbers, we don't need to use four as a building block. Four is already been built up into a block. What about if we introduced five. So if we had fives, we can now have one times five. Just let me highlight here. We can have one times five couldn't we? We could have two times five, we could have three times five. Now we said we don't have worry about fours, so we could do two times two times five to get 20. Now we've got five, we can do five times five. We can't do five times six, because we haven't said we can use sixes but six is made up of two times three. So we can do two times three times five Five times seven. We can't make that, that's not going to be possible. Because there's no way that we can make seven, we've got the five, but seven is also a prime number. And since I haven't said you can use sevens yet, we can't make 35. Is there anything else that we can make while we're at it, we could do two times two times two times five to get 40. And We could do three times three to get nine and three times three times five will be 45. And we could do two times five times five to get 50. So using ones, twos, threes and fives, we can make all of these numbers. What about if we tried six? Does it matter? Well, six is not prime, is it? Six is made up of two times three. So we don't need to introduce six into here to be able to get it. If we introduce six, we're not going to make any new numbers. The next new number that we need to introduce is seven. And once again, seven is prime. And those prime numbers are really important to everything we do today. Two, three, five and seven, they are our first prime numbers. Let's move on to the next bit. So the grid shows all the numbers that can be made by multiplying twos, threes and fives. Can you tell me how we made 27? Have a think about the numbers that we used. We did three, multiplied by three, multiplied by three. Don't forget we said we can only use two, three or five. What about 40? How did we make 40? Well, if you're not quite sure Let's have a think, 40 could be made by doing four times 10. Unfortunately, we can't do four times 10, can we?. It could be however, four is made of two times two. And 10 is made up of two times five. So that's the way that we could have gotten there, isn't it? So now's your turn. Pause the video and access the worksheet, have a go at the questions and when you're ready, un-pause it and we can go through it together, good luck. How did you get on? Let's go through the answers together. Question one says complete the frames find ways of showing the numbers as products of different combinations of factors. So there's more than one answer here. You might get a different one to me. 24 could be made up of four times six. Now, if we look at three factors, well four is made up of two times two, isn't it? Two times, two times six. That's another way to make 24. We could break that six down into two times three. Making that two times two there, so there's some answers, you might have got. 120, it ends in a zero so it's a multiple of 10. That can be 10 times 12. And 10 is made up of two multiplied by five, I'm just going to multiply by 12 and I keep two times five, and I could say that 12 is two times six. And that's fine. That's a set of answers that matches that question. Now, one thing to bear in mind is that when we came down to this bottom row here, in this case, all the numbers were prime . Whereas in this case, we still got a number that's not prime. Nothing wrong with that for that answer. It didn't ask for prime factors. Question two does, complete the frames So each equation shows a number as a product of its prime factors. So we're only allowed to use prime numbers to make this work. So question a 12 is two times three times something two threes is six, six times two makes 12, doesn't it? 32 times something times five, two times five is 10. So that's going to be three. I've just noticed this went down rather than across right? 45 is three times something times something. Well, how many threes make 45? 11 threes make sorry, 15 times three is 45. So 15 is three times five. Let's go across. So 20, two times five is 10. So that must be two. Two times two is four times...yep. 36, three times three is nine. Nine times two is 18, so I need to multiply that by another two to get 36. 54, two times three is six. Now six times nine makes 54, nine is a product of it's prime factors and nine's made up of three times three. Okay, let's have a look on the right hand side now. Well, to fill in the squares, the numbers have been cut out from 100 Square, and they've been rubbed out, we're going to use the clues to writing the correct numbers in the grids. So we know five squared is 25. And six squared is 36. We are about to fill in the rest of the numbers in that grid. So 24, 25 we've got, 26, 27, 28 underneath that we're increasing by 10. So we've got 34, 35, 36, 37, 38. Okay, let's go down. So three lots of four squared, four squared is 16 and three lots of that will be 48. Two cubed, multiplied by seven, two cubes two times two times two is eight and eight times seven is 56. Two times five squared, five squared is 25. Two lots of that is 50. And now we need to fill in the gaps between 40 and 50. That must be 49, 46 and 47, 57, 58, 59 and 60. Okay, last one is a little bit tricky. Three cubed, multiply by two. Well, three cubed is 27. And 27 times two is 54. Four to the power of three, that's four times four times four, four times four is 16, 16, 32, 48, 64. And that's good as well because directly underneath that should work. Two cubed is eight. Three squared is nine. That makes 72. Three to the power of four, three times three times three times three. Three times three is nine. Three times three is nine, if I multiply those together I'll get 81. And now we can fill in the gaps. 53, 52, 51, 61, 62 and 63, 71 73, 74, 82, 83 and 84. So we'll finish with the Explore activity. Who do you agree with? And can you explain your answer? So Anthony says the last digit of two to the power of 100 is zero. And Ben thinks that it can't be that. But do you think the last digit is? Pause the video and have a go. When you're ready, un-pause it and we'll do it together. pause in 3, 2, 1. Right, I'm going to tell you now I'm not going to work out the value of 2 to a 100. It means when two times two times two times two times two is going to go on and on. It's a number that's going to be too big for me certainly for me to do in my head. I don't think it's something that my calculator could do. So we're not going to work out the value of the whole number. We're only interested in that last digit. Let's start off with what we do know. Two to the power of one, it's two. In fact you know what? I don't want to do that there, I want some space that I can go down the page that I'm writing. Two to the power of one is two. Two to the power of two, two times two is four. Two to the power of three is eight. Two to the power of four, 16. We're doubling every time. Two to the power of five, 16. Double this 32. Two to the power of six, if we double that we get 64 I'm just going to go a little bit further, I don't want to go on forever. 128, two to the power of eight, 256. Let's just stop there just for a moment. And let's look at those last digits each time. We've got two, four, eight, six, two, four, eight, six, two, four, eight, six. So I can predict, now I'm going to predict two to the power of nine is going to end at two. Now, if we double turn 256, we get 512. So we've got a pattern that we can see. Let's go sketch a table. So, two to the power of one. So the first one ends in two. The second one ends in four. Third one ends in an eight. The fourth one ends in a six. The fifth one ends in a two. But we're trying to work out the last digit of the hundredth one, aren't we? So can we predict it from this? So the sixth one, ends in a four. Seventh one will end in an eight. The eighth one, will end in six. Then the ninth one is going to end in a two. And the tenth one, will end in a four. So we need to look at this and see if we can work out what 100th one is going to be. Now it repeats every fourth time 24862486. So that means, if we do 100 divided by four, we get 25. So it's going to repeat this same sequence 25 times. Which is the fourth one ends in a six and the eighth one ends in a six. And the 12th one, let's draw a new table, let's have a different colour. So I'm just going to rub that out to get some space in there. Make sure that I get this right. So we're saying the 12th one, ends in a six. So the 16th one, will end in a six. 20th will end in a six, the 24th will end in a six. The 28th will end in a six, the 32nd will end in a six. The 36th will end in a six, the 40th will end in a six. I won't keep following this up because you can see if the 20th ends in a six, and the 40th ends in a six, I can be pretty certain, that the 60th will end in a six. We can see that pattern, the 80th will end in a six, and so the 100th one of the sequence is also going to end in a six. That's it for this lesson today. I'll see you later. Goodbye.

Prime Building Blocks

Prime Building Blocks

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