**Least Common Multiple and Highest Common Factor **

** Introduction: **

** * Factor and Multiple: **If the first number of two numbers completely divide the Second Number, then the first number is **Factor **of the second number And the second number is called **Multiple** of the first number.

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

*** Least common Multiple (L.C.M.):** The small number that is divided completely from each given number, the given numbers are called the Least Common multiple.

** * Methods to Calculate L.C.M.: **The following two methods to determine L.C.M are: –

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

**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

L.C.M. = 2

Here, the biggest deficit of 2 and 3 is 2

** (2) [Division Method] **

**Method –
(i) **put the given numbers in a row and divide by the lowest number of prime numbers 2 3 5 7 11 etc., which will be completely divided by at least two of these given numbers.

**(ii) **After that, the remainder which is received and those numbers which can not be divided 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 any action.

**(iv) **Multiply all the divisors and the last row numbers and get the 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 –**To find the L.C.M. of given decimal numbers Find the LCM of their corresponding full number and put the decimal after the number in the LCM, as the number of decimal points in the minimum digits from the right side.

** 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 in number 2.4 the number of digits after decimal is one. That is why L.C.M. of full numbers 36 after put a decimal in 360 after the ‘0’ from the right side.

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

** Formula: **

**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 HCF of two or more numbers (H.C.F.) is the largest number that divides each of them completely.

**The method of finding the greatest common divisor – **The following two methods of computing the greatest common divisors are:

**1. Prime Factorization Method
**

**Example: **What will be the HCF 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 common divisor of 493 and 928?

Solution:

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

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

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

**Method – **

To find the H.C.F. of given decimal numbers Find the H.C.F of their corresponding full number and put the decimal after the number in the H.C.F, as the number of decimal points in the maximum digits from the right side.

**Example: **What will be the greatest common divisor of 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

**H.C.F. of fractions **

** Formula: **

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

**TYPE-1
**

**Example 1:** What would be the greatest common divisor 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 common divisor?

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

** TYPE – 2
**

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

** 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 divisor and the LCM two numbers are 12 and 396, respectively. If one of them is 36, what would be the second number?

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