### LCM and HCF

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__H.C.F and L.C.M__

Highest Common Factor (H.C.F)

H.C.F is the highest common factor or also known as greatest common divisor, the greatest number which exactly divides all the given numbers.

There are two methods to find H.C.F of given numbers, they are:

i. Prime factorization method.

ii. Division Method.

ii. Division Method.

**HCF of given fractions**

H.C.F of given fractions =

L.C.M of denominator

H.C.F of numerator

Least Common Multiple (L.C.M)

L.C.M is least common multiple, the smallest number which is exactly divisible by all the given
numbers.

There are two methods to find L.C.M of given numbers, they are:

There are two methods to find L.C.M of given numbers, they are:

i. Prime factorization method.

ii. Division Method.

ii. Division Method.

**LCM of given fractions**
L.C.M of given fractions =

H.C.F of denominator

L.C.M of numerator

Important Points

1. Product of two numbers = H.C.F *L.C.M of the two numbers

2. The smallest number which when divided by x, y and z leaves a remainder R in each case.

3. The greatest number which divides x, y and z to leave the remainder R is H.C.F of (x - R), (y - R) and (z - R)

4. The greatest number which divide x, y, z to leave remainders a, b, c is H.C.F of (x - a), (y - b) and (z - c)

5. The smallest number which when divided by x, y and z leaves remainder of a, b, c (x - a), (y - b), (z - c) are multiples of K

6. The LCM will be the product of multiplying the highest power of each prime number together.example: lcm(8,9,21) ,The highest power of the three prime numbers 2, 3, and 7 is 2

7. The HCF can be computed by determining the prime factorizations of the two numbers and comparing factors, example:hcf(18, 84), prime factors 18 = 2 * 3

2. The smallest number which when divided by x, y and z leaves a remainder R in each case.

Required number = (L.C.M of x, y, z) + R

3. The greatest number which divides x, y and z to leave the remainder R is H.C.F of (x - R), (y - R) and (z - R)

4. The greatest number which divide x, y, z to leave remainders a, b, c is H.C.F of (x - a), (y - b) and (z - c)

5. The smallest number which when divided by x, y and z leaves remainder of a, b, c (x - a), (y - b), (z - c) are multiples of K

Required number = (L.C.M of x, y and z) - K

6. The LCM will be the product of multiplying the highest power of each prime number together.example: lcm(8,9,21) ,The highest power of the three prime numbers 2, 3, and 7 is 2

^{3}, 3^{2}, and 7^{1}, respectively. Thus,lcm = 8*9*7 =5047. The HCF can be computed by determining the prime factorizations of the two numbers and comparing factors, example:hcf(18, 84), prime factors 18 = 2 * 3

^{2}and 84 = 2^{2}* 3 * 7 and notice that the "overlap" of the two expressions is 2 * 3; so hcf = 6.