**Least Common Multiple and Highest Common Factor **

** Introduction: **

** * Factor and Multiple: **If the first number of two numbers completely divided, the first number of the second number is **Factor** And the second number is called **Multiple** of the first number.

** Example: **Number 3 is a factor of 24, whereas number 24 is the number of number 3.

*** Shortest Complexity (L.C.M.):** The small number that is divided completely from each given number, the given numbers are called the shortest metabolism.

** * Shortest Closure Removal: **The following two methods to determine L.C.M are: –

**(1) [Prime Factorization Method] **

**Verb method –
(i)** First of all, express each of the given numbers as a prime factor.

** Example: **What is L.C.M. of 18, 28, 108 and 105?**
Here, **18 = 2 × 3 × 3 = 2 × 3

28 = 2 × 2 × 7 = 2

108 = 2 × 2 × 3 × 3 × 3 = 2

And 105 = 3 × 5 × 7

Distressed L.C.M. = 2

Here, the biggest deficit of 2 and 3 is 2

** (2) [Division Method] **

** verb method –
(i) **put the given numbers in a row and divide the lowest number of prime numbers 2 3 5 7 11 etc., which will be completed in at least two of these given numbers. Run away

**(ii) **After that, the gaps which are received and those numbers which can not be distributed in the given numbers, fall in the second row.

**(iii) **The first line of action can be done in the second, third rows with the smallest number, it may be possible to repeat an action.

**(iv) **Multiply all the divisors and the last row numbers and get the received L. C. M.

**Example: **What is L.C.M. of 36, 60, 84 and 90?

** Solution: **

Therefore, the specified L.C.M. = 2 × 2 × 3 × 3 × 5 × 7 = 1260 ** Ans.**

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

** verb method –**L.C.M. of given decimal numbers To find their corresponding full number of LC.M. Remove and get the decimal number inserted after the number in the LCM by number, as the number of decimal points in the decimal is decimal to the right after the decimal.

** Example: **What is L.C.M. of 2.4, 0.36, 0.045?

** Solution: **

L.C.M. of compatible full number = 3 × 3 × 4 × 2 × 5 = 360

So, nowadays L.C.M. = 36.0 = 36 ** Ans.**

**Note:** Here number 2 is the number of digits after decimal in decimal number. That is why L.C.M. of full numbers A decimal is set in 360 after the ‘0’ from the right side.

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

** Formula: **

**Minimal wings of exposure and exponent (L.C.M. of Power and Base)
**

**TYPE-1
**

**Example 1. **3^{7 } 3^{12} 3^{17} What will be the smallest endowment?

** Solution: **LC.M. = 3^{17} ** Answer. **

**Example 2.** 5^{-9} 5^{-7} 5^{-14} What will be the smallest endowment?

** Solution: **5^{-7}** Answer. [∵ -7 > -14]**

**TYPE – 2
**

**Highest Common Factor:** The greatest converter of two or more numbers (H.C.F.) is a large number that divides each of them completely.

**The method of finding the greatest convergent – **The following two methods of removing the greatest particle are:

**1. Prime Factorization Method
**

**Example: **What will be the greatest consumer of 28 and 32?**
Solution: **28 = 2 × 2 × 7 and

32 = 2 × 2 × 2 × 2 × 2

The given numbers 28 and 32 have the common factor 2 and 2, so the desired H.C.F = 2 × 2 = 4

** 2. Continued Division Method
**

**Example: **What will be the greatest exponent of 493 and 928?

Solution:

**H.C.F. of three or more numbers Detection Method:**

** Action Method –
(i) **The first two numbers of H.C.F. Determine by continuously splitting method,

**H.C.F. of Decimals**

** Action Method – **

H.C.F. To find their corresponding full numbers, H.C.F. Remove and get H.C.F. Put the decimal after the number of points on the right side, as the number of maximum digits in decimal is numbered after the decimal point,

**Example: **What will be the greatest contributor to 1.5, 0.24 and 0.036?**
Solution: **15 = 3 × 5

24 = 3 × 2

36 = 3 × 3 × 2

H.C.F = 3 of corresponding full

Therefore, the desired H.C.F = 0.003

**The greatest Fractions of fractions **

** Formula: **

**The greatest exponent of power and exponent (H.C.F. of Power and Base)
**

**TYPE-1
**

**Example 1:** What would be the greatest consumer of 2^{8} 2^{10} 2^{15} ?

** Solution: **Expected H.C.F = 2^{8 }

**Example 2:** 7^{-12 } 7^{-13} 7^{-18} What would be the greatest consumer?

** Solution: **Expected H.C.F = 7^{-18}** Answer. [∵ -18 < -12]**

** TYPE – 2
**

** Example 1:** 5^{2 } 4^{3} What will be the greatest consumer?

** Solution: **5^{2} = 1 × 5^{2} and 4^{3} = 1 × 4^{3}

Expected H.C.F = 1** Answer. **

** Formula: First Number × Second Number = H.C.F × L.C.F**

** Example: **The greatest common denominator of the two numbers and the smallest clusters are 12 and 396, respectively. If one of them is 36, what would be the second number?

Follow