In the view of competition, finding problems on HCF and LCM needs you to be much actuated. Keeping an eye on each factor and making out the major dissimilarity between two concepts is really hard over and above puzzling task. So friends! Let’s clear both the concepts and make them on your tips to perform best in your examinations.

**Before going further take a look at:**

__Test of Divisibility__

__Test of Divisibility__:

**Divisibility by 2 or 5**: A number is divisible by 2 or 5 if the unit digit is divisible by 2 or 5.**Divisibility by 3**: A number is divisible by 3 if the sum of its digit is divisible by 3.*(For example 695421, here sum of all digits=27 which is divisible by 3)***Divisibility by 4**: A number is divisible by 4 if the last two digits are divisible by 4*(For example 6879376 is divisible by 4 as last digits 76 is divisible by 4)***Divisibility by 6**: A number is divisible by 6 if it is divisible by both 2 and 3.*(For example 24 divisible by both 2 and 3 both)***Divisibility by 8**: A number is divisible by 8 if the number formed by hundred’s, ten’s and unit’s digit of the number is divisible by 8.*(For example 16789352 is divisible by 8 as number formed by last three-digit i.e. 352 is dividable by 8)***Divisibility by 9**: A number is divisible by 9 if the sum of its digits is divisible by 9.*(For example 246591 is divisible by 9 as the sum of its digits is 27 divisible by 9)***Divisibility by 10**: A number is divisible by 10 only when a unit digit is 0.*(For example 3200 is completely divisible by 3)***Divisibility by 11**:A number is divisible by 11 if the difference between the sun of the digit at odd places and at even places is either 0 or divisible by 11.*( For example 29435417 is divisible by 11 as (7+4+3+9)-(1+5+4+2)=23-12=11 is divisible by 11)***Divisibility by 7,11,13**: A number is divisible by 7,11 and 13 if and only if the difference of its thousand and reminder is divisible by 7,11 and 13*( For example 473312 is divisible by7 since 473-312=161 is divisible by 7)***Divisibility by 12**: All the numbers divisible by 3 and 4 both are divisible by 12.

**HCF or GCM or GCD**

Highest common Factor or Greatest Common Multiple or Greatest Common Divisor of two numbers is the greatest number that divides each of them exactly.

#### Methods To Find HCF:

*Method 1: Finding The Highest Factor*

Take an example: **Find the HCF of 8, 12, and 24**

**Step 1:** Split number into their individual factors as:

**Step2:** Find the common factor among factors of three numbers.

**Step3:** 4 is the highest common factor among factors of three numbers.

Hence the HCF of 8, 12 and 24 is 4.

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*Method 2 :Factorization:*

Take the same example:

**Step1:** Find the multiples of a given number:

*8= 2x2x2*

*12=2x2x3*

*24=2x2x2x3 *

**Step2:** Then express the given number as prime factors

*8= 2x2x2 or you can write 2 ^{3}x3^{0}*

*12=2x2x3 or 2 ^{2}x3^{1}*

*24=2x2x2x3 or 2 ^{3}x3^{1}*

**Step3: **Find the product of least power of common prime factors

*2 ^{2}x3^{0} = 4 is the least power and is the HCF of 8, 12, and 24 *

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*Method 3: Division Method*

Take the same example: **Step1:** Divide larger number by smaller number and then divide the divisor by remainder:

**Step2:** Repeat the process till 0 is obtained as the remainder.

**Step3: **The last divisor is required HCF i.e. 4

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**Lowest Common Multiple or Lowest Common Multiple (LCM)**

The least number which is exactly divisible to each one is called their LCM.

#### Methods To Find LCM:

*Method 1: Factorization Method*

Take an example:

**Find the LCM of 72,108 and 2100**

**Step1:** Resolve the numbers into prime factors:

*72= 2x2x2x3x3 or 2 ^{3}x3^{2}*

*108=3x3x3x2x2 or 3 ^{3}x2^{2}2*

*100=2x2x5x5x3x7 or 2 ^{2}x5^{2}x 3 x 7*

**Step2: **LCM is the product of the highest powers of all the factors:

**LCM= 2 ^{3}x3^{3}x5^{2}x7 =37800 **

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*Method 2: Common Division Method(Short Cut Method):*

Take an example: **Find the LCM of 16, 24, 36 and 54**

**Step1:** Divide the numbers till 1 is obtained:

**Step2: **Find the LCM by multiplying the divisors:

**LCM= 2x2x2x3x3x3x2=432**

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**** HCF and LCM Of fractions****

HCF= HCF of Numerator/LCM of Denominator

LCM= LCM of Numerator/HCF of Denominator

*IMPORTANT FACTS:*

- Product of two number= Product of HCF X LCM
- If HCF of x & y is A then HCF of x, (x+y) is also G.
- If HCF of x & y is A then HCF of x, (x-y) is also G.
- If HCF of x & y is A then HCF of (x-y), (x+y) is also G.

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