H.C.F. & L.C.M. OF NUMBERS
 
If a number x divides another number y exactly, then x is said to be a factor of y. To add, subtract, or just compare two fractions, we need to convert both fractions to a common denominator. The least common multiple is usually used, although any common multiple would work in this particular case. Here comes the concept of Least Common Multiple (L.C.M.). Also to simplify a fraction, we divide the numerator and the denominator by the same number. If we divide them by the greatest common factor, then no further simplifications are required. That greatest common factor is called the Highest Common Factor (H.C.F.) or Greatest Common Divisor (G.C.D.). Least Common Multiple (L.C.M.): The least number which is exactly divisible by each one of the given numbers is called their L.C.M. There are two methods of finding the L.C.M .of two or more than two numbers:
1. Factorization Method: In this method first resolve each one of the given numbers into a product of Prime factors. Then Least Common Multiple is the product of highest powers of all the factors.
2. Common Division Method: In this method first arrange the set of numbers in any order. Then take a number which divides exactly at least two of the given numbers and carry forward the remaining numbers which are not divisible by that particular number. Repeat the process till no two of the numbers are divisible by the same number. Then Least Common Factor is the product of the divisors and the undivided numbers.
Highest Common Factor (H.C.F.): The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly. There are two methods of finding the H.C.F. of two or more than two numbers: Factorization Method: First resolve each one of the given numbers as the product prime factors. Then H.C.F. is the product of least powers of common prime factors. Division Method: If we want to find out the H.C.F. of two numbers then divide the larger number by smaller one. Again divide the divisor by the remainder. We should repeat this process till zero is obtained as remainder. If we have three numbers then H.C.F. of {(H.C.F. of any two) and (the third number)} will be the H.C.F. of three given numbers. Important Formulas of L.C.M. and H.C.F. of Numbers – Product of two numbers =Product of their H.C.F and L.C.M.
1. H.C.F. = H.C.F.of NumeratorsL.C.M.of Denominators
2. L.C.M. = L.C.M.of NumeratorsH.C.F.of Denominators
Solved Examples 
Example 1. Find the L.C.M. of 50, 35 and 70. Solution. Let us apply the Factorization Method to find the L.C.M. of 50, 35 and 70. Express each one of the given number as the product of prime factors.
50 = 2 × 52, 35 = 5 × 7, 70 = 2 × 5 × 7. ∴L.C.M. = Product of highest powers of 2, 5 and 7 = 2 × 52 × 7 = 350. 
Example 2. Find the L.C.M. of 18, 20, 24, and 54. Solution. Let us apply Common Division Method to find the L.C.M. of 18, 20, 24, 54.

Directions: Mark the correct Answer:
1. Find the highest common factor of 54 and 36.
(a) 4
(b) 6
(c) 16
(d) 18

2. Find the lowest common multiple of 24, 20 and 30.
(a) 120
(b) 50
(c) 64
(d) 24

3. The H.C.F. of 25 , 169 , 1825 is:
(a) 4225
(b) 2225
(c) 24125
(d) 2125

4. The L.C.M. of 23 , 1627 , 512 is:
(a) 803
(b) 809
(c) 695
(d) 495

5. The ratio of two numbers is 3:4 and their H.C.F. is 4.Their L.C.M. is:
(a) 12
(b) 16
(c) 24
(d) 48

6. The L.C.M. of two numbers is 189 and H.C.F. of two numbers is 9. If one of the number is 63 then other number is:
(a) 27
(b) 28
(c) 29
(d) 30

7. The H.C.F. of 3.6, 7.2, 2.1, 0.81 is:
(a) 0.3
(b) 0.03
(c) 0.9
(d) 0.09

8. The L.C.M. of 1.8, 3.6, and 0.72 is:
(a) 3.6
(b) 11.8
(c) 0.36
(d) 10.80

9. Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case
(a) 4
(b) 7
(c) 9
(d) 13

10. Three different containers contain 204 litres, 1190 litres and 1445 litres of Pepsi, Slice and Coca-Cola respectively. What biggest measure can be measured from all the different quantities of cold drinks exactly?
(a) 17
(b) 18
(c) 19
(d) 20

Answers
1. (d)
2. (a)
3. (b)
4. (a)
5. (d)
6. (a)
7. (b)
8. (d)
9. (a)
10. (a)
Solution and Explanation
1. 54 = 2×3×3×3, 36 = 3 ×2×3×2,
Therefore, H.C.F. = 2×3×3 =18.
2. 24 = 2×2 ×2×3, 20 = 2×2 ×5 , 30= 2×3×5.
Therefore, L.C.M. = products of highest powers of 2, 3 and 5 = 23 × 3 × 5 =120.
8. L.C.M. of 180, 360,72 =1080.
Therefore, L.C.M. of 1.8, 3.6, and 0.72 is 10.80
9. Required number will be = H.C.F. of (91 -43), (183 -91) and (183 – 43) = H.C.F. of 48, 92, 140 =4