Calculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1


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1 Calculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1 What are the multiples of 5? The multiples are in the five times table What are the factors of 90? Each of these is a pair of factors. There are 6 pairs of factors, hence 1 factors. 1 5 = 5 5 = 10 5 = = = = 90 0 = = = = 90 the number itself is a multiple. So 5, 10, 15, 0, etc are multiples of 5 1 and the number itself are also factors. So 1,,, 5, 6, 9, 10, 15, 18, 0, 45, 90 are factors of 90 Lowest Common Multiples Calculate the Lowest Common Multiple of 45 and 60. Multiples of 45: 45, 90, 15, 180, 5, Multiples of 60: 60, 10, 180, 40, It s 180 a) Calculate the Lowest Common Multiple of 8 and 1. List the multiples of the two numbers & ask 'what is the smallest number common to both lists?' This is the LCM. So 180 is the LCM of 45 and 60 Definition The Lowest Common Multiple of or more numbers is the lowest number that can be divided by all of these numbers. Highest Common Factors Calculate the Highest Common Factor of 45 and 60. Factors of 45: 1,, 5, 9, 15, 45. Factors of 60: 1,,, 4, 5, 6, 10, 1, 15, 0, 0, 60. b) Calculate the Highest Common Factor of 8 and 1. It s 15 List the factors of the two numbers & ask 'what is the largest number common to both lists?' This is the HCF. So 15 is the HCF of 45 and 60 Definition The Highest Common Factor of or more numbers is the highest number that can divide into all of these numbers. Observation 180 is the Lowest Common Multiple of 45 and is the Highest Common Factor of 45 and 60. The Lowest Common Multiple is a BIG number The Highest Common Factor is a SMALL number 1. Calculate the LCM of 1 and 0.. Calculate the HCF of 1 and 0.
2 Know what a Prime Number is NA A prime number is special because it only has TWO factors: 1 and itself. Here is a list of factors for the first few whole numbers. 1: 1 : 1,. : 1,. 4: 1,, 4. 5: 1, 5. 6: 1,,, 6. 7: 1, 7. Are either of 101 or 1001 prime? Ask the question Try the next prime Does go into 101 exactly? Does go into 101 exactly? Does 5 go into 101 exactly? Does 7 go into 101 exactly? Does 1 go into 101 exactly? You can keep trying but you will find no additional factors! Which numbers have exactly TWO factors? The first seven prime numbers are:,, 5, 7, 11, 1, 17, 1 is not a prime number as it has only ONE factor Look for the factors systematically by dividing the number using the prime numbers Prime Numbers Does go into 1001 exactly? Does go into 1001 exactly? Does 5 go into 1001 exactly? Does 7 go into 1001 exactly? If you find another factor stop! 1001 is NOT prime because it has at 101 is prime. least three factors i.e. 1, 1001 and 7! a) Blank out the above before you start; now list the numbers 1 to 5 and decide which are prime. Extra When do you stop trying to find factors? Stop looking when you have tried all of the prime numbers up to the square root of the number. Continuing from the example of whether 101 is a prime number, 1 1 is bigger than 101 so there is no factor between 1 and 101 i.e There is no additional factor between 1 and 101, so there are no additional factors. If there is an additional factor pair, then the smallest factor of the pair must be between 1 and the square root of the number. Hey Why stop there? It is based on the idea that factors come in pairs AND (in this example) that the missing factor pair lies between and Write Whole Numbers as the Product of Primes NA Write 90 as the product of primes. Remember Product means multiplication 90 does divide by exactly! Method Draw a factor tree So 90 written as a product of primes: 90 = 5 b) Write 160 as the product of primes Begin by dividing by the first prime number,. If the number does not divide by this prime, then try dividing by the next prime and then the next etc. 45 does not divide by, so try This is 90 written as its product of prime factors 1. What is a prime number?. Which of, 19, 77, 100 are prime?. Write 100 as the product of primes.
3 Calculate with Negative Numbers with & without a Calculator using + NA Two negatives multiplied together are positive. Similarly, two negatives one divided by the other are positive. The results are summarised as follows: 1. 4 = 8. 4 =. 4 = = + + = = + = + = + + = + = + = + = Also use the rules when + symbols are next to each other! 5. or written ( ) = + = or written + ( ) = = 0 Find a) 8 b) 1 9 c) 10 Hint: This is a way of writing10 d) 8 4 e) 5 6 Calculator Check Make sure you can use your calculator each one is different Do the examples 14 above on your calculator. Check you get the 4 results shown. To enter a negative number 6 OR 1 ± 6 ±. use the ± button or the ( ) button, e.g. To enter 1 6, press, ( ) 1 ( ) Reminder Normal subtractions are calculated by simply moving along the number line. 4 = = Find Substitute Numbers into a Formula NA4 1 Replace the letter, x, with For the equation, y = x + 7, its specified value,. For the equation, y = x 7x +, find y when x is. It s an excellent habit to place the find y when x is substituted values in brackets!! Look at the sign to the y = ( ) + 7 Reminder left of the number to help y = ( ) 7( ) = + you do the multiplication. y = = + y = = + = 7 ( ) = 14 y = 1 y = = 8 Remember ( ) = ( ) ( ) = 4 a) In s 1) and ) above, find y when x is. b) In s 1) and ) above, find y when x is. 1. Find y when x is, where y = 7 x. Find y when x is 5, where y = 1 x
4 Know & use the Index Laws (using numbers) NA5 Index Laws 0 = 1 All numbers to the power 0 are 1 a b = a+b For multiplication just add the indices a b = a b For division just subtract the indices ( a ) b = a b For power (repeated multiplication) just multiply the indices These are the index laws using the number. The laws work with any number as long as the base number is the same! So 4 a 4 b = 4 a+b But watch out with 4 a 5 b. Why do the index laws work? Calculate 4 4 means The base numbers are different so index laws don t work!! means So 4 means which is just 6 5 means 5 which means. Cancel the s, = = This is just the two indices added i.e. + 4 This is just the two indices subtracted i.e. 5 ( 5 ) means 5 5 5, this means ( ) ( ) ( ) = 15 We could just add these indices = 15!! This is just the two indices, 5 and, multiplied i.e. 5 a) Learn the four index laws above b) Write i) 6 in the form a ii) (4 4 ) 6 in the form 4 b Index laws Know & use the Index Laws (using letters) NA5 a 0 = 1 All numbers to the power 0 are 1 a x a y = a x+y For multiplication just add the indices a x a y = a x y x a For division just subtract the indices also written a y = a x y (a x ) y = a xy For power (repeated multiplication) just multiply the indices These laws are essentially the same as above but having replaced the base number with the letter a a. a a 9 = a + 9 = a 1 b. b 4 b = b 4 = b 1 = b The letters must be the same for the index laws to work! c. (c 4 ) 7 = c 4 7 = c 8 c) Learn these four index laws d) Simplify i) e 5 e ii) (m 8 ) 4 Remember Numbers to the power 1 are just themselves e.g. 1 = OR b 1 = b Simplify using the index laws ( 8 ) 7 4. x x 9 5. y 5 y 6. (z 4 ) 7
5 Round to a Given Number of Significant Figures (s.f.) NA6 ignore these! Round to 1 s.f The number after the required significant figure is 1. So round down. Round down by ignoring numbers after. So this becomes just to 1 s.f.!! 1 s.f. just means the first digit which isn t a zero, i.e. in this case. Round to 5 s.f This is the 5th significant figure. The number after this is 9. This is 5 or higher so round up. Round up by increasing this by one. So this becomes.1417 to 5 s.f.!! 6+1 = 7 Look at the number after the required significant figure. Round up if this number is 5 or higher. Round down if this number is 4 or below. Round down by ignoring numbers after the required significant figure. Round up by rounding the required significant figure up by one digit and ignore numbers thereafter. Round to a) s.f. b) s.f. c) 4 s.f. Round to s.f. This is the rd significant figure. The number after this is 5. This is 5 and over so round up. Round up by increasing this by one. So this becomes 0.41 to s.f.!! For SMALL numbers between 0 and 1 Watch Out!! Front noughts are not significant = 1 This is not a front nought so it is significant This is a front nought so it is not significant! d) Round to s.f. e) Round to 4 s.f. Hint: these noughts are not significant. These ones still are significant Extra notes on Estimation In your examination you will be expected to round your answers as appropriate, often to or significant figures. Occasional examination questions will require you to estimate a simple numerical calculation. Normally this requires you to simply round the numbers to a convenient number and then calculate  often rounding to 1 significant figure is appropriate Estimate = = 1. 5 Estimate (. 9) 8 ( 4). 15 = = 16 Note: means approximately equals to Round the following to the number of significant figures in brackets () () () ()
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