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# LCM and HCF Formulas

## Basic Formulas of LCM and HCF

LCM and HCF formulas, properties and important ways to solve LCM and HCF Questions are given below on this page below.

## Formulas of LCM and HCF

**Property 1 –** LCM x HCF = Product of two numbers

**H.C.F. and L.C.M. of Fractions:**

1. H.C.F. = | H.C.F. of Numerators |

L.C.M. of Denominators |

2. L.C.M. = | L.C.M. of Numerators |

H.C.F. of Denominators |

**Property 2 –** LCM ≥ Numbers ≥ HCF

**Property 3-** LCM is a multiple of HCF

**Property 4 –** If the HCF of two numbers is 1 then they are Co-Primes.

How to calculate LCM and HCF, is given below later on this page.

**What are Multiples –**

Multiple is a series of numbers that are exactly divisible by a particular number. e.g. 8, 12, 16, 20 …. are the multiples of 4.

**What are Factors –**

Factors for a given number are the list of numbers that would divide the larger number without leaving any remainder ( Except 1 and the number itself). For Example – for 12 the factors would be 2, 3, 4, 6

For 6 – 12, 18, 24, 30 will be called as multiple of 6.

For 24 – 6 and 12 are Factors of 24.

**What is Lowest Common Multiple (LCM) –**

As the name suggests LCM is the lowest common multiple of two or more Natural Numbers for e.g. for 15 and 20, 60 is LCM(Don’t worry we will explain how we calculated this)

**What is Highest Common Factor (HCF) or Greatest Common Divisor (GCD) –**

The Largest(Highest) common Factor of two or more numbers will be called as HCF of the number. e.g. for 12 and 15. 3 will be the HCF.

**Prime Factors –**

These are unique list prime numbers that divide the greater number e.g. for 20 – 2, 5 are the prime Factors.

### Calculating Prime Factors –

In this method, you take the lowest prime number and see if the greater number is divisible by it. If its not divisible then you move to the higher prime number.

Let us show you how to calculate the prime factors for number step by step.

Prime Factors of 12 –

- Lowest Prime number 2, 12 ÷ 2 = 6
- Again 6 ÷ 2 = 3
- Again 3 ÷ 2 is not possible so next prime number is 3
- Again 3 ÷ 3 = 1

We can write this as 12 = 2x2x3 = 2^{2} × 3

**Finding LCM of two Numbers –**

Let us take an example HCF of 15 and 20, first we list out all the prime factors of each

15 = 3 x 5

20 = 2 x 2 x 5 = 2^{2} × 5

Then multiply each factor by the greatest number of times it occurs in either number.

- 2 – Occurs 2 times
- 3 – Occurs 1 time
- 5 – Occurs 1 time

So LCM is (2×2)x(3×1)x(5×1) = 60

**Finding HCF of two Numbers**

Let us take an example HCF of 18 and 24, we already have listed out all the prime factors of each number

18 = 2 x 3 x 3

24 = 2 x 2 x 2 x 3 = 2^{3} × 3

Now we find the factor that exists at least once in both of them.

### Using Formulas of HCF and LCM in Questions

**Question: Calculate the highest number that will divide 43, 91 and 183 and leaves the same remainder in each case**

**Options**

**A. 4**

**B. 7**

**C. 9**

**D. 13**

**Solution: **Required number = H.C.F. of (91 – 43), (183 – 91) and (183 – 43)

H.C.F. of 48, 92 and 140 = 4.

**Correct Answer : A**

**Question: Which of the following is greatest number of four digits which is divisible by 15, 25, 40 and 75 is:**

**Options**

**A. 9700**

**B. 9600**

**C. 9800**

**D. 9650**

**Solution: ** Greatest number of 4-digits is 9999.

Now , find the L.C.M. of 15, 25, 40 and 75 i.e. 600.

On dividing 9999 by 600, the remainder is 399.

Hence, Required number (9999 – 399) = 9600.

**Correct Answer : B**

**Question: The greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm, 12 m 95 cm is:**

**Options**

**A. 25 cm**

**B. 15 cm**

**C. 35 cm**

**D. 55 cm**

**Solution:** Required length = H.C.F. of 700 cm, 385 cm and 1295 cm = 35 cm.

**Correct Answer : C**

**Read Also – How To Solve HCF and LCM Problems Quickly**

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