FACTORS AND MULTIPLES Answer Key


 Ethelbert Norman Charles
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1 I. Find prime factors by factor tree method FACTORS AND MULTIPLES Answer Key a = 2*2*2*2*2*2*2*2 *3 b = 3*2*2*2*67 c = 5*3*163
2 II. Find prime factors by prime factorization method a Ans = 3* 931 b Ans = 5*7* 17 c Ans = 7*17 III. Find if the following are coprime or not. If they are not coprime find a common factor. a. 486, 81 = These numbers are not coprime as they have a common factor which is 3.(3 divides both 486,81 without a remainder) b. 49,59 = These numbers are coprime as they don t have a common factor which divides both numbers without a remainder c. 75,101 = These numbers are coprime as they don t have a common factor which divides both numbers without a remainder d. 144,192 = These numbers are not coprime as they have two common factors which are 2, 3. (2 and 3 divides both 144,192 without a remainder) e. 246,780 = These numbers are not coprime as they have a common factor which is 2.(2 divides both 246,780 without a remainder)
3 f. 634,597 = These numbers are coprime as they don t have common factor which divides both numbers without a remainder IV. Find the LCM of a , 80, , 16, 5 2 7, 16, 1 2 7, 8, 1 2 7, 4, 1 * 7, 2, 1 Ans = 5*5*2*2*2*7*2 = 2800 b , 200, , 100, , 50, , 25, , 25, 4 3 9, 25, 2 5 3, 25, 2 3, 5, 2 * Ans = 2*2*2*2*2*3*5*3*5*2 = c , 48, , 48, 5 3 9, 16, 5 2 3, 16, 5 2 3, 8, 5 2 3, 4, 5 3, 2, 5 *
4 Ans = 5*3*3*2*2*2*3*2*5 = 10,800 d , 66, , 22, , 11, , 11, 1 7, 11, 1 Ans = 3*2*2*2*7*11*1 = 1848 e , 25, , 5, , 1, , 1, 14 11, 1, 2 * Ans = 5*5*2*7*11*2 = 7700 f , 330, , 66, , 33, , 33, 4 2 2, 11, 4 1, 11, 2 * Ans = 5*2*7*3*2*11*2 = 9240 g. 5 65, , , 4 1, 2 Ans = 5*13*2*2= 260
5 V. Find the HCF by prime factorization method a. 3 27, 36, , 12, 6 3, 4, 2 Ans = 3*3 = 9 b. 5 25, 95, , 19, 16 Ans = 5*5 = 25 c , 500, , 100, , 20, , 4, 2 3, 2, 1 Ans = 5*5*5*2 = 250 d , , , , 7 2, 1 Ans =2*3*3*7 = 126 e. 2 56, 144, , 72, 54 14, 36, 27
6 Ans =2*2 = 4 VI. Find the HCF by successive division method a. 106,192, ) 192( ) 106( ) 86( ) 20(318 2) 6(36 0 HCF of 106 and 192 is 2 2)96(48 _96_ 0 Ans  HCF of 106,192 and 96 is 2 b. 108,144,60 108) 144(1 _ ) 108(3 _ So, HCF of 108 and 144 is 36
7 36) 60( ) 36(1 _24 12)24( Ans HCF of 108,144,60 is 12 c. 7, 14,24 7) 14 ( So, HCF of 7,14 is 7 7) 24( Ans HCF of 7,14,24 is 7 d. 32,56,46 32)56( ) 32( ) 24(324 0
8 So HCF of 32 and 56 is 8 8) 46(540 6) 8 (16 2)6(36 0 Ans = HCF of 32,56,46 is 2 VII. Fill in the blanks a. Every number is a factor of itself b. The only even prime is 2 c. All even numbers are multiples of 2 d. The 12 th multiple of 5 is 60 e. Numbers which have only 1 and itself as factors are called prime numbers f. Numbers which have more than one factor are called composite numbers g. Two numbers are coprime if they don t have a common factor h. Factors of 81 are 9 and 9 i. Set of whole numbers is { 0,1,2,3,4..} j. 563,291,497,893 are odd numbers k. Two numbers are not coprime if they have a common factor l. Multiple of 7 and 8 is 56 m. The smallest multiple of every number is itself n. A number that divides another number without a remainder is called its factor ***************** THE END*********************
9 Summary Factors and Multiples A number which divides another number without leaving a remainder is called its factor. Eg. 3 divides 27 without a remainder. So 3 is a factor of 27 A number is a multiple of another number if it is divisible by it Set of Natural numbers { 1,2,3,4.} Set of Whole numbers { 0,1,2,3,4..} Set of Integers {3,2,1,0,1,2,3 } Even numbers end with 0,2,4,6,8 and all even numbers are multiples of 2 Odd numbers end with 1,3,5,7,9 A number which had only one and itself as its factors is called prime numbers Numbers which have more than one factor is called composite 1 is unique and is neither prime nor composite Only even prime is 2 Two numbers which have no factors in common are called coprime Two numbers which have factors in common are not coprime
10 1. Finding factors by factor tree method Summary HCF and LCM Since it is called factor tree, use the tree method(this clue will help you not to get confused with the prime factorization method since it looks like a tree) Divide the given number using only prime numbers till you get two prime numbers in the end Then answer should be written as Given number = Product of all the prime factors at each step eg Answer 1608 = 3*2*2*2*67 Use only prime numbers 2. Finding factors by prime factorization Divide the number using only prime numbers till you cannot divide any further Answer should be written as Product of all the factors and the numbers in the last row Use only prime numbers Ans = 5*7* LCM Divide all the given numbers using only prime numbers. You can use prime numbers that need not divide all the given numbers. In that case, carry down the numbers which are not divisible as it is Continue till you can t divide further Answer = Product of all factors and the numbers on the last row 5 65, , , 4 1, 2 Number brought down as it is since it is not divisible by 5 Always use prime numbers
11 Ans = 5*13*2*2= HCF Prime Factorization method Divide all the given numbers using a prime number. Note that here the prime number must divide all the given numbers unlike LCM Continue till you can t find a prime number that will divide all the numbers Answer = Product of all factors 2 56, 144, , 72, 54 14, 36, 27 Stop division since there is no prime which divides all 3 numbers Ans =2*2 = 4 Note: In order to differentiate between LCM and HCF just remember In LCM take the factors on the vertical and horizontal (like in the letter L). L In HCF take only the factors in the vertical In LCM, prime numbers need not divide all given numbers but HCF it must divide all the numbers 5. HCF Successive division method Remainder becomes second divisor Eg HCF of 108,144 and ) 144(1 _ ) 108(3 _ Divide second number by the first This number becomes the second dividend Continue the same process till you get zero The last divisor is the HCF for the first two numbers
12 Then divide the third number with this HCF in the same manner till you get zero. The last divisor is the HCF of all 3 numbers Divide the third number by the HCF of the first numbers 36) 60( ) 36(124 Last divisor is the HCF of all three numbers 12)24( Ans HCF of 108,144,60 is 12 ******************** THE END***********************
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