# LCM and HCF Shortcuts, Formulas and Tricks

## LCM and HCF Shortcuts, Formulas and Tricks

Hello
Friends, Today we are sharing **LCM and
HCF Shortcuts, Formulas and Tricks PDF.** LCM and HCF section is very
important part of all competitive exams.
It is very helpful for various type of exam like SSC CGL, CHSL, RRB,
Insurance, Bank exams and other competitive exams.

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**Problems on H.C.F and L.C.M - Important
Formulas**, H.C.F and L.C.M Important Formulas.

**Highest Common Factor (HCF)**

Factor:
Numbers which are multiplied together to get another number is called factors
of that number. e.g. 3 and 4 are factors of 12, because 3 × 4 = 12. A number
can have many factors.

Common
factors: Factors that are common to two or more numbers are called common
factors.

Highest
Common Factor: HCF of two or more numbers is the greatest number that divides
each of them exactly

**Method I:**

*Factorisation
or prime numbers method of finding HCF :*

Step 1:
Express each number as a product of prime factors.

Step 2:
HCF is the product of all common prime factors using the least power of each
common prime factor.

e.g.

Find the
HCF of 27 and 36

27 = 3^{3}

36 = 3^{2 }x 2^{2}

Find the
HCF of 108, 288 and 360.

108
= 2^{2} x 3^{3}

288
= 2^{5} x 3^{2}

360
= 2^{3} x 5 x 3^{2}

HCF
= 2^{2} x 3^{2 }= 36

**Method II:**

### Division
Method to find HCF (Shortcut):

Step 1: Divide the larger number by the smaller number

Step 2: Divisor of step 1 is divided by its
remainder

Step 3: Divisor of step 2 is divided by its remainder. Continue
this process till we get 0 as the remainder.

Step 4: Divisor of the last step is the HCF.

e.g. Find the HCF of 108 and 288.

**How to find HCF of three numbers using division method**

Step 1:
Find out HCF of any two numbers.

Step 2:
Find out the HCF of the third number and the HCF obtained in step 1

Step 3:
HCF obtained in step 2 will be the HCF of the three numbers

Note: The
Number that divides each of the two numbers also divide their sum, their
difference and the sum and difference of any multiples of that numbers.

e.g HCF of 42 and 70

= HCF of
42 and 28(70-42)

= HCF of
28 and 14 (42-28)

= HCF of
14 and 14 (28-14)

= 14,

Hence, HCF
= 14

(keep this
method in mind, it gives result very quickly)

**HCF of decimals:**

Make the
decimal place same in all the given numbers, find the HCF as they are integers,
and adjust decimal accordingly in the final result.

e.g Find
the HCF of 16.5, 0.45 and 15.

Solution:

numbers
can be written as, 16.50, 0.45 and 15.00

now find
the HCF of 1650, 45 and 1500, which is 15.

Adjust the
decimal accordingly, i.e. 0.15

**HCF of Fractions: **

The **HCF** of two or more fractions is the
highest fraction which is exactly divisible by each of the fractions.

**HCF = (HCF of Numerators) / (LCM of denominators)**

**Least Common Multiple (LCM)**

**Multiple:** Multiple is the product of any
quantity and an integer.

**Common Multiple:** Common
multiple is that which is multiple of two or more numbers. The common multiples
of 2 and 3 are 0, 6,12,18, 24, etc.

**Least Common Multiple (LCM)** of two or
more numbers is the smallest number (other than zero) that is a multiple of all
the numbers

**Example**: LCM of 5 and 6 = 30 because 30 is
the smallest multiple which is common to 5 and 6 (Or we can say that 30 is
the smallest number which is divisible by both 5 and 6)

**How to
find LCM of Given numbers:
Method I:**

__Factorisation
or prime numbers method to Find LCM:__

Step 1:
Express each number as a product of prime factors.

Step 2: LCM is the product of highest powers of all prime factors

Example 1: Find out LCM of 8 and 14

Step 1: Express each number as a product of prime factors.

8 = 2^{3}

14 = 2 × 7

Step 2: LCM is the product of highest powers of all prime factors

Here the prime factors are 2 and 7

The highest power of 2 here = 2^{3}

The highest power of 7 here = 7

Hence LCM = 2^{3} × 7 = 56

**Method II:**

*Division
Method to Find LCM (Shortcut):*

Step 1:
Write the given numbers in a horizontal line separated by commas.

Step 2: Divide the given numbers by the smallest prime number which can exactly
divide at least two of the given numbers.

Step 3: Write the quotients and undivided numbers in a line below the first.

Step 4: Repeat the process until we reach a stage where no prime factor is
common to any two numbers in the row.

Step 5: LCM is the product of all the divisors and the numbers in the last
line.

e.g. Find the LCM of 20, 25 and 30 by long division method.

**LCM of
decimals:**

Make the
decimal place same in all the given numbers, find the LCM as they are
integers and adjust decimal accordingly in the final result.

e.g. Find
the LCM of 0.6, 9.6 and 0.36

Solution:

numbers
can be written as, 0.60, 9.60 and .36

now find
the LCM of 60, 960 and 36, which is 2880.

Adjust the decimal accordingly, i.e. 28.80

**LCM of
Fractions: **

The LCM of
two or more fractions is the least fraction or integer which is exactly
divisible by each of them.

**LCM = (LCM of Numerators) / (HCF of
denominators)**

Note:

*Product of two numbers = Product of their HCF and LCM**HCF of numbers always divides LCM exactly***.**

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