# Calculating Lowest Common Multiple (LCM)

**Method 1:**

**Note:**Donâ€™t repeat any factor while writing the product in step2.

**Example:**Find the L.C.M. of 36, 56, 105 and 108.

- 36 = 2
^{2}Ã— 3^{2}; 56 = 2^{3}Ã— 7; 105 = 3 Ã— 5 Ã— 7; 108 = 2^{2}Ã— 3^{3} - The L.C.M. must contain every prime factor of each of the numbers. Also it must include the highest power of each prime factor which appears in any of them. So, it must contain 2 or it would not be a multiple of 56, it must contain 3 or it would not be a multiple of 108, it must contain 5 or it would not be a multiple of 105, and it must contain 7 or it would not be a multiple of 56 or of 105. Therefore, the L.C.M. = 2
^{3}Ã— 3^{3}Ã— 5 Ã— 7 = 7560

**Method 2:**

Write the factor on the left hand side which can divide maximum of the numbers.

Multiply all the factors or divisors and the numbers left in the last row. The product gives the L.C.M. of the given numbers.

**L.C.M. of Decimals**

Example

Find out the L.C.M. of 0.16, 5.4, and 0.0098.

Solution

First of all we find out the L.C.M. of 16, 54, 98. Here L.C.M. of 16, 54, 98 is 21168. In numbers 0.16, 5.4, 0.0098, the minimum digits from right to left is 5.4. Here 5.4 the decimals is given of one digit from right to left. So, we put decimal in our result such that: 21168 = 1226.8

Example

Find the L.C.M. of 48, 10.8 and 0.140.

Solution

L.C.M. of 48, 108 and 140 is 15120. So, L.C.M. of 48, 10.8 and 0.140 = 1.5120

**L.C.M. of Fractions**

=

# Calculating Highest Common Factor (HCF)

**Method 1 (Factorization Method)**

**Factor method has discussed above or, we express each given number as the product of primes. Now we take the product of common factors which is our required H.C.F.**

**Example:**Find the H.C.F of 144, 336 and 2016.

**Method 2 (Division Method)**

**We divide the greater number by the smaller number and find out the remainder.**

If a/b, c/d, e/f be the proper fractions then their H.C.F. is given by

# Important Tips

- Product of two numbers = L.C.M. Ã— H.C.F.
- Product of n numbers = L.C.M of n numbers Ã— Product of the HCF of each possible pair
- If ratio of numbers is a : b and H is the HCF of the numbers Then LCM of the numbers = H x a x b = HCF x Product of the ratios.
- H.C.F of fractions = H.C.F. of Numerators L.C.M. of Denominators
- L.C.M. of fractions = L.C.M. of Numerators H.C.F. of Denominators
- If HCF (a, b) = H1 and HCF (c, d) = H2, then HCF (a, b, c, d) = HCF (H1, H2).
- LCM is always a multiple of HCF of the numbers . i.e. LCM = (a number) x HCF
- To find the GREATEST NUMBER that will exactly divide x, y, z.
- To find the GREATEST NUMBER that will divide x, y and z leaving remainders a, b and c respectively.
- To find the LEAST NUMBER which is exactly divisible by x, y and z.
- To find the LEAST NUMBER -which when divided by x, y and z leaves the remainders a, b and c respectively.
- Find the LEAST NUMBER, which when divided by x, y and z leaves the same remainder â€˜râ€™ each case. (L.C.M. of x, y and z) + r.
- Find the GREATEST NUMBER that will divide x, y and z leaving the same remainder in each case.
- or TWO NUMBERS: FIRST NUMBER * SECOND NUMBER = L.C.M * H.C.F
- FIRST NUMBER =
- SECOND NUMBER =
- LCM =
- HCF =

**Rule of finding Dividend**

Dividend = Divisor* Quotient + Remainder |