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ASSIGNMENT SUB : Mathematics CLASS : VI a WEEK – 8 ………………………………………………………………………………………………………… ASSIGNMENT SUB : Mathematics CLASS : VI A WEEK – 8 ………………………………………………………………………………………………………… Chapter :- 03 , Factors and Multiples Topics :- (i) Highest common factor (HCF) (ii) Least common Multiples (LCM) (iii) Relation between HCF and LCM _____________________________________________________________________________________ i. Highest common factor (HCF) The Highest common factor (HCF) of two or more numbers is the largest numbers in the common factor of the numbers. HCF is also known as GCD (Greatest common Divisor) The following are the methods of finding the HCF of two or more numbers. i) Finding HCF by listing factors. ii) Prime Factorisation Method iii) Short Division Method iv) Continued or long Division Method. 1. Finding HCF by listing factors :- Consider three numbers 36, 42 and 24. Factors of 36 = 1 2 3 4 6 9 12 18 36 Factors of 42 = 1 2 3 6 7 14 21 42 Factors of 24 = 1 2 3 4 6 8 12 24 Common factors (HCF) of 36 , 42 and 24 = 1, 2, 3, 6 HCF = 6 which exactly divides the numbers 36, 42 and 24. 2. Prime factorization method :- In this method first find the prime factorization of the given numbers and then multiply the common factors to get HCF of the given numbers. Prime factors of 48 = 2 x 2 x 2 x 2 x 3 Prime factors of 96 = 2 x 2 x 2 x 2 x 3 x 2 Prime factors of 144 = 2 x 2 x 2 x 2 x 3 x 3 Common factors = 2 , 2 , 2 , 2 , 3 HCF of 48 , 96 and 144 = 2 x 2 x 2 x 2 x 3 = 48 Ans. 3. Short Division Method :- Divide all the numbers by a common factor of all and continue the process till there is no common factor for all of them. Example :- Find the HCF 32, 216 , 144. Solution :- Common factors of 32, 216 and 144 = 2 , 2 , 2 So, the HCF of 32, 216 and 144 = 2 x 2 x 2 = 8 Ans. 4. Continued (or long division) method. Step 1 :- Divide the bigger number by the smaller number. Step 2 :- If the remainder is zero, then the smaller number is the HCF of the numbers. If there is a remainder, then consider this remainder as new divisor and then divide the smaller number by the new divisor. Step 3 :- Repeat this process until there is no remainder. The last divisor is the required HCF of the given numbers. Example :- Find the HCF of 144 , 252 and 330 by continued division method. Solution :- First we find the HCF of 144 and 25 So, the HCF of 144 , 252 and 330 is 6 Ans. Ex. 3.3 Q1. Find the HCF of the following set of the numbers by listing factors method. a. 144 , 180 c. 136 , 170 , 255 Q2. Find the HCF of prime factorization method. a. 304 , 180 c. 524 , 128 , 56 Q3. Find the HCF by long division method. a. 175 , 625 d. 110 , 770 , 1331 Q4. Find the HCF by short division method. a. 85 , 136 d. 70 , 105 , 175 , 65 Q6. The length , breadth and height of a room are 8 m 50 cm, 5 m 75 cm and 1 m 75 cm respectively. Find the length of the longest tape which can measure the three dimensions of the room. [Hint. Find the HCF of 850 cm, 575 cm and 175 cm. Least common Multiples (LCM) The least common multiples (LCM) of the two or more numbers is the number which is least (or lowest) among all the common multiples. There are three methods for finding LCM. i) Finding LCM by listing the multiples. ii) Prime factorization method. iii) Common Division Method. iv) Finding LCM by listing the Multiples. Consider two numbers 4 and 6. Multiples of 4 = 4 8 12 16 20 24 28 32 36 40 Multiples of 6 = 6 12 18 24 30 36 42 Common Multiples of 4 and 6 = 12 , 24 , 36 ……………… In these common multiples, we see that 12 is the least. So the LCM of 4 and 6 = 12 2. Prime Factorisation Method Step – 1 :- Find the prime factorization of the given numbers. Step – 2 :- Find the common factors. Step – 3 :- Multiply the common factors and the other factors also, that is common and un common factors are to be multiplied. Example :- Find the LCM of 48, 72 and 120 Solution :- The prime factorization on the given number are :- So, 48 = 2 x 2 x 2 x 2 x 3 72 = 2 x 2 x 2 x 3 x 3 120 = 2 x 2 x 2 x 5 x 3 LCM = Common x Uncommon numbers = 2 x 2 x 2 x 3 x 2 x 3 x 5 = 720 LCM of 48 , 72 and 120 = 720 3. Common division method. Step 1 :- Write the given numbers in a rows separated by commas. Step 2 :- Divide these numbers by the least prime number which divides at least one of the given numbers. Step 3 :- Write the quotients and the numbers that are not divisible by the prime number in the second row. Then, repeat Step 2 and Step 3 with the rows and continue till the numbers in a row are equal to 1. Step 4 :- The LCM is the product of all the prime divisors. Examples :- Find the LCM of 32, 48 and 96. Solution :- The LCM of 32 , 48 and 96 is the product of the all the divisors. So, the LCM of 32 , 48 and 96 = 2 x 2 x 2 x 2 x 2 x 3 = 96 Ans. Ex. 3.4 Q1. Write all natural numbers less than 90 which are common multiples of 3 and 5. Q2. Find the LCM by the Prime factorization method. a) 120 , 250 c. 216 , 108 , 36 Q3. Find the LCM by the common division method. a) 135 , 90 d. 60 , 40 , 32 , 80 Q4. Find the smallest number which is exactly divisible by :- a) 32 , 72 b. 20 , 65 , 90 Q5. a) Find the smallest number which when divided by 24 , 36 and 64 leaving 4 as remainder each case. [Hint :- Find LCM of 24 , 36 and 64 and add 4] b) A number is divisible by 24 , 25 and 120 if it is increased by 20. Find the number. [Hint :- Find the LCM of 24 , 25 and 120 and subtract 20] 3. Relation between HCF and LCM. i. Product of given numbers :- HCF x LCM Product of given numbers or HCF = LCM Product of given numbers or LCM = HCF Consider two numbers 12 and 54 12 = 2 x 2 x 3 54 = 2 x 3 x 3 x 3 HCF = 2 x 3 = 6 LCM = 2 x 3 x 2 x 3 x 3 = 108 Product of given numbers = HCF x LCM 12 x 54 = 6 x 108 648 = 648 proved Exercise 3.5 Q2. The product of two numbers is 2625. If the HCF of the numbers is 5 , find the LCM of the numbers. Q4. If the HCF and LCM of two numbers are 12 and 72 respectively, find the product of the two numbers. Q5. The LCM of two numbers is 819. If the two numbers are 63 and 117, find the HCF. Q7. The HCF and LCM of the two numbers are 5 and 400 respectively. If one of the numbers is 25, find the other number. Solution of 01 Given – HCF = 5 LCM = 400 One number = 25 by using formula 16 HCF x LCM 5 x 400 Other Number = = One number 25 = 80 Ans. Note :- Solve all the questions of ex. 3.3 , 3.4 and 3.5 .
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