India
CHAPTER 2 L.C.M. AND H.C.F.
1 Common Factor A common factor of two or more numbers is a number that divides each of them exactly. Ex. Let the numbers be 27 and 81. In this 3 is common factor. 2. Common Multiple A common multiple of two or more numbers is a number which is exactly divisible by each of them. Ex. 12 is common multiple of numbers 4 and 6. 3. L.C.M. L.C.M. stands for least common multiple. L.C.M. of two or more given numbers is the least number which is exactly divisible by each one of them. Thus, 20 is a common multiple of 4 and 5. 40 is a common multiple of 4 and 5. 60 is a common multiple of 4 and 5. But 20 is the least common multiple (L.C.M.) of 4 and 5 2 1, 2, 6 3
L.C.M of 8, 24 is 2 × 2 × 2 × 3 × 1 = 24. Ex. 3. Find the L.C.M of 12, 15, 90, 108, 135 and 150. Sol. 2 3 3 5 12, 15, 90, 108, 135, 150 45, 54, 135, 75 18, 45, 25 6, 15, 25 6, 3, 5 • • • • ... (ii) ... (iii) ... (iv) ... (v) ... (i)
in row (i) 12 and 15 are factors of 108 and 90 respectively. Hence we struck off 12 and 15. in row (ii) 45 is factor of 135, therefore 45 is struck off. in row (v) 3 is factor of 6, therefore 3 is struck off. The required L.C.M. = 2 × 3 × 3 × 5 × 6 × 5 = 2700.
Methods of finding L.C.M Method-I. By Prime factorization
Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of the highest powers of all the factors. Ex. 1. Find the L.C.M. of 72, 108 and 2100. Sol. 72= 2 × 2 × 2 × 3 × 3 = 23 × 32 108= 3 × 3 × 3 × 2 × 2 = 33 × 22 2100= 2 × 2 × 5 × 5 × 3 × 7 = 22 × 52 × 3 × 7 L.C.M. of 72, 108 and 2100 = Product of highest powers of 2, 3, 5 and 7. L.C.M. of 72, 108 and 2100 = 23 × 33 × 52 × 7 = 37800.
4.H.C.F. H.C.F. stands for Highest Common Factor or Greatest Common Multliple, i.e., G.C.M. H.C.F. of two or more numbers is the greatest number that divides each of them exactly. Thus, 7 is the H.C.F. of 21 and 28. Because there is no number greater than 7 that divides both 21 and 28. ... (1) Methods of
1 Common Factor A common factor of two or more numbers is a number that divides each of them exactly. Ex. Let the numbers be 27 and 81. In this 3 is common factor. 2. Common Multiple A common multiple of two or more numbers is a number which is exactly divisible by each of them. Ex. 12 is common multiple of numbers 4 and 6. 3. L.C.M. L.C.M. stands for least common multiple. L.C.M. of two or more given numbers is the least number which is exactly divisible by each one of them. Thus, 20 is a common multiple of 4 and 5. 40 is a common multiple of 4 and 5. 60 is a common multiple of 4 and 5. But 20 is the least common multiple (L.C.M.) of 4 and 5 2 1, 2, 6 3
L.C.M of 8, 24 is 2 × 2 × 2 × 3 × 1 = 24. Ex. 3. Find the L.C.M of 12, 15, 90, 108, 135 and 150. Sol. 2 3 3 5 12, 15, 90, 108, 135, 150 45, 54, 135, 75 18, 45, 25 6, 15, 25 6, 3, 5 • • • • ... (ii) ... (iii) ... (iv) ... (v) ... (i)
in row (i) 12 and 15 are factors of 108 and 90 respectively. Hence we struck off 12 and 15. in row (ii) 45 is factor of 135, therefore 45 is struck off. in row (v) 3 is factor of 6, therefore 3 is struck off. The required L.C.M. = 2 × 3 × 3 × 5 × 6 × 5 = 2700.
Methods of finding L.C.M Method-I. By Prime factorization
Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of the highest powers of all the factors. Ex. 1. Find the L.C.M. of 72, 108 and 2100. Sol. 72= 2 × 2 × 2 × 3 × 3 = 23 × 32 108= 3 × 3 × 3 × 2 × 2 = 33 × 22 2100= 2 × 2 × 5 × 5 × 3 × 7 = 22 × 52 × 3 × 7 L.C.M. of 72, 108 and 2100 = Product of highest powers of 2, 3, 5 and 7. L.C.M. of 72, 108 and 2100 = 23 × 33 × 52 × 7 = 37800.
4.H.C.F. H.C.F. stands for Highest Common Factor or Greatest Common Multliple, i.e., G.C.M. H.C.F. of two or more numbers is the greatest number that divides each of them exactly. Thus, 7 is the H.C.F. of 21 and 28. Because there is no number greater than 7 that divides both 21 and 28. ... (1) Methods of