Table of contents 
HCF: Highest Common Factor 
LCM: Least Common Multiple 
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(b) Division Method
(a) Factorisation Method
Example 1: Find the L.C.M of 2, 4, 6, 8, 10.
 Write the numbers as the product of primes.
 2, 2^{2}, 2× 3, 2^{3}, 2× 5
 Take the highest powers of all the primes. i.e., 23 × 3 × 5 = 120.
Example 2: Find the H.C.F. of 2, 4, 6, 8, 10 using the factorization method.
 For H.C.F, take the common prime factors. i.e., 2. (This is only the prime factor common in all the numbers).
(i) Product of two numbers = L.C.M. × H.C.F.
(ii) Product of n numbers = L.C.M of n numbers × Product of the HCF of each possible pair.
Or
If the HCF of all the possible pairs taken is same then we will have:
Product of n numbers = L.C.M of n numbers × (H.C.F of each pair)^{(n – 1)}
(iii) If the ratio of numbers is a : b and H is the HCF of the numbers. Then,
LCM of the numbers = H × a × b = HCF × Product of the ratios.
(vi) H.C.F of fractions =
(vii) L.C.M. of fractions =
(viii) If HCF (a, b) = H_{1} and HCF (c, d) = H_{2}, then HCF (a, b, c, d) = HCF (H_{1}, H_{2}).
Example 3: What is the greatest number which exactly divides 110, 154 and 242?
 The required number is the HCF of 110, 154 & 242.
 110 = 2 × 5 × 11
 154 = 2 × 7 × 11
 242 = 2 × 11 × 11
 ∴ HCF = 2 × 11 = 22
Example 4: What is the greatest number, which when divides 3 consecutive odd numbers produces a remainder of 1.
 If x, y, z be 3 consecutive odd numbers, then the required number will be the HCF of x – 1, y – 1 and z – 1.
 Since x1, y1 & z1 are 3 consecutive even integers, their HCF will be 2. So the answer is 2.
Example 5: What is the highest 3 digit number, which is exactly divisible by 3, 5, 6 and 7?
 The least no. which is exactly divisible by 3, 5, 6, & 7 is LCM (3, 5, 6, 7) = 210. So, all the multiples of 210 will be exactly divisible by 3, 5, 6 and 7.
 So, such greatest 3 digit number is 840 = (210 × 4).
Example 6: In a farewell party, some students are giving pose for photographs, if the students stand at 4 students per row, 2 students will be left if they stand 5 per row, 3 will be left and if they stand 6 per row 4 will be left. If the total number of students are greater than 100 and less than 150, how many students are there?
 If ‘N’ is the number of students, it is clear from the question that if N is divided by 4, 5, and 6, it produces a remainder of 2, 3, & 4 respectively.
 Since (4 – 2) = (5 – 3) = (6 – 4) = 2, the least possible value of N is LCM (4, 5, 6) – 2 = 60 – 2, = 58.
 But, 100 < N < 150. So, the next possible value is 58 + 60 = 118.
Example 7: There are some students in the class. Mr X brought 130 chocolates and distributed to the students equally; then he was left with some chocolates. Mr Y brought 170 chocolates and distributed equally to the students. He was also left with the same no of chocolates as MrX was left. Mr Z brought 250 chocolates, did the same thing and left with the same no of chocolates. What is the max possible no of students that were in the class?
 The question can be stated as, what is the highest number, which divides 130, 170 and 250 gives the same remainder, i.e. HCF ((170 −130),(250 −170),(250 −130)).
i.e. HCF (40, 80, 120) = 40.
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