**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.

(ii) Select the biggest power from all of these undivided factors which is included in the multiplication of any number among them.

(iii) Now Find out the LCM by multiplying the undivided Factors.

Example: What is L.C.M. of 18, 28, 108 and 105?

Here, 18 = 2 × 3 × 3 = 2 × 3^{2}

28 = 2 × 2 × 7 = 2^{2} × 7

108 = 2 × 2 × 3 × 3 × 3 = 2^{2} × 3 ^{ 3}

And 105 = 3 × 5 × 7

L.C.M. = 2^{2} × 3^{3} × 5 × 7 = 4 × 27 × 5 × 7 = 3780 Ans.

Here, the biggest deficit of 2 and 3 is 2^{2} and 3^{3} respectively, and the second prime factor is 5 and 7 respectively.

** (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.

**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.

Formula:

TYPE-1

Trick – If the base number of the given numbers is same and the power is different, then their LCM is the maximum power number

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

TRICK – If the base number and the power of the given numbers is different, then their L.C.M. Factor is computed by Multiplication method.

**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**

Action Method –

(i) Express each of the given numbers as a prime factor.

(ii) Find out the product of numbers which is present in all numbers i.e., the most, the product of these numbers will be HCF.

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 Ans.

** 2. Continued Division Method**

Action Method –

(i) First of all In the given numbers, divide the largest numbers by the smallest numbers.

(ii) Then divide the remaining ones by the divisor. this is the second divisor.

(iii) If still left remainder then it will again divided by another divisor.

(iv) repeat this action until the remainder is zero.

(v) Thus, the lastest divisor is the given H.C.F.

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.

(ii) Then find the H.C.F of this H.C.F and remaining Numbers.

(iii) Perform this method with all the given numbers.

(iv) Then the Last H.C.F. will be final H.C.F.

**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^{3}

36 = 3 × 3 × 2^{2}

H.C.F = 3 of corresponding full

Therefore, the desired H.C.F = 0.003 Answer.

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

Formula:

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

TYPE-1

TRICK – If the base of the given numbers is identical and the power is different, then their H.C.F. is the number with minimum power number?

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

TRICK – If the base and power of the given numbers is different, then their H.C.F. is calculated by Factorization Method.

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?

Follow