L.C.M and H.C.F

L.C.M and H.C.F

Introduction

The concepts of Least Common Multiple (L.C.M) and Highest Common Factor (H.C.F) are essential in mathematics, particularly in number theory. L.C.M is particularly useful when dealing with fractions, while H.C.F is crucial when simplifying fractions.

Definitions

Least Common Multiple (L.C.M)

The Least Common Multiple of two or more numbers is the smallest multiple that is exactly divisible by each of the numbers. For example, the L.C.M of 4 and 5 is 20.

The L.C.M can be found using the formula:

L.C.M of a a and b b = a×bH.C.F(a,b)\frac{a \times b}{\text{H.C.F}(a, b)}

Highest Common Factor (H.C.F)

The Highest Common Factor of two or more numbers is the largest number that divides each of the numbers without leaving a remainder. For example, the H.C.F of 12 and 15 is 3.

H.C.F of a a and b b = Greatest factor that divides both a a and b b .

Example Calculations

Finding L.C.M

Calculate the L.C.M of 8 and 12.

  • Factors of 8: 1, 2, 4, 8
  • Factors of 12: 1, 2, 3, 4, 6, 12

The common multiples of 8 and 12 are 24, 48, 72... Therefore, the L.C.M of 8 and 12 is 24.

Finding H.C.F

Calculate the H.C.F of 24 and 36.

  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

The common factors are 1, 2, 3, 4, 6, and 12. The highest of these is 12. Therefore, H.C.F of 24 and 36 is 12.

Word Problems

Problem 1

Sam has 18 apples, and he wants to pack them in boxes of 6. How many boxes can he fill?
Solution: H.C.F of 18 and 6 is 6. Sam can fill 3 boxes.

Problem 2 (Involving Roman Numerals)

A theater has 48 seats in one row and 60 seats in another row. If they want to arrange the chairs in such a way that each row has the same number of chairs with no chairs left over, what is the maximum number of chairs in each row that can be used?
Solution: H.C.F of 48 and 60 is XII \text{XII} (12). Thus, a maximum of 12 chairs can be used in each row.

Conclusion

Understanding L.C.M and H.C.F enhances our ability to work with numbers, particularly in solving real-life problems like arranging items and simplifying fractions. Both concepts are integral in further mathematics and daily calculations.