**HCF and LCM:** Basic Arithmetic plays a vital role in laying a strong foundation of Mathematics, a subject that is dreaded by many. With your concepts clear right from the beginning, you won’t face any difficulty in understanding the various advanced concepts that are taught in higher classes. Highest Common Factor and Lowest Common Multiple HCF and LCM are two such concepts that find importance not only for school-level Mathematics but also in various other exams, like CAT, MAT, recruitment exams for government jobs, etc. It is, therefore, important that you understand the Properties if HCF and LCM properly.

In this article, we will explain what are HCF and LCM, their definition, the relationship between HCF and LCM, properties of HCF and LCM, along with examples.

## HCF And LCM

Let us first look at the definition of each:

### What Is HCF?

HCF stands for **Highest Common Factor**. It is also known as GCF (Greatest Common Factor) or GCD (Greatest Common Divisor).

Factors of a number are numbers that divide the given number without leaving any remainder. That is they are exact divisors of the given number. For example, the number 12 is divisible by 2, 3, 4, and 6. These are all factors of the number 12.

The Highest Common Factor of two or more numbers is the highest number by which all the given numbers are divisible without leaving any remainders. Basically, it is the largest number that divides all the given numbers.

### HCF Example

Let’s consider two numbers – 12 and 18.

As mentioned above, the factors of 12 are 1, 2, 3, 4, 6, and 12.

On the other hand, the factors of 18 are 1, 2, 3, 6, 9, and 18.

Both 12 and 18 have 1, 2, 3, and 6 as common factors, of which 6 is the highest. So, 6 is the HCF of 12 and 18.

### What Is LCM?

LCM stands for Lowest or Least Common Multiple. The LCM of two or more numbers is the smallest positive integer that is divisible by all the given numbers.

### LCM Example

Let us consider two numbers – 8 and 12.

The multiples of 8 are: 8 X 1 = **8**, 8 X 2 = **16**, 8 X 3 = **24**, 8 X 4 = **32**, and so on…

The multiples of 12 are: 12 X 1 = **12**, 12 X 2 = **24**, 12 X 3 = **36**, 12 X 4 = **48**, and so on…

Of all these multiples of 8n and 12, 24 is the lowest and common multiple of both. So, 24 is the LCM of 8 and 12.

### HCF And LCM Formulas, Tricks: How To Find HCF And LCM?

There are various methods to calculate HCF and LCM.

However, the easiest way to calculate LCM and HCF is the division method that uses the common prime factors. Let us explain this method with an example:

Let us consider two numbers 18 and 32.

First, let us find the common prime factors of these two numbers.

### How To Find Common Prime Factors?

The steps to find the common prime factors of given numbers are as under:

**1st Step:** Write the given numbers horizontally, separated by a comma. Draw a line under the two numbers.

**2nd Step:** On the left side, draw a horizontal line and then write the smallest prime number that divides the given numbers without leaving any remainder. Elaborating with our example of 8 and 20:

**3rd Step:** Now, divide both numbers by the lowest prime number and write the quotients, separated by a comma, under the horizontal line.

**4th Step:** Repeat Step 2 and 3 until you reach the stage wherein no common prime factor is available.

**5th Step:** The numbers on the left-hand side (inside the red box) are the common prime factors of the given numbers.

### Calculating HCF And LCM

**How To Find HCF**?

Now, to calculate HCF, find the product of the common prime factors of the numbers.

In our case, the HCF of 8 and 12 is: 2 X 2 = 4.

**How To Find LCM?**

To calculate the LCM, we can use the common prime factors found using the above method.

First, we need to write the prime factors of each of the given numbers. In our case, referring to the image above:

8 = 2 X 2 X 220 = 2 X 2 X 5 |

Now, we have to find the maximum occurrences of each of these factors in any number.

Referring to our example of 8 and 20:

The factor, 2 occurs the most for 8 = 3 times. The factor, 5 occurs the most for 20 = 1 time. |

Now, LCM can be calculated by following these two steps:

a. Multiply each of the factors the maximum number of times it occurs:

2 occurs three times; so, 2 X 2 X 2 = 8. 5 occurs one time; so, 5 X 1 = 5. |

b. Multiply the resultant numbers to get the LCM. In our case:

LCM = 8 X 5 -= 40 |

### Properties Of HCF And LCM

Some of the important properties of HCF and LCM are as under:

a. The HCF of given numbers is never greater or more than any of the numbers.

b. The LCM of given numbers is never less than any of the numbers.

c. The HCF of two or more prime numbers is always 1.

d. The LCM of two or more prime numbers is their product.

e. The product of two numbers, a and b, is equal to the product of their HCF and LCM. This means:

a X b = LCM of (a & b) X HCF of (a & b) |

f. LCM of Fractions = \(\frac{LCM of Numerators}{HCF of Denominators}\)

g. HCF of Fractions = \(\frac{HCF of Numerators}{LCM of Denominators}\)

### HCF And LCM Examples

Here are a few examples of HCF and LCM:

*Example 1: Find the HCF and LCM of 15 and 24.*

*Example 1: Find the HCF and LCM of 15 and 24.*

Let us first find the prime factors of 15 and 24:

15 = 3 X 5 24 = 2 X 2 X 2 X 3 |

As 3 is the only factor that is common for both 15 and 24, HCF = 3.

Now, to find LCM of 15 and 24:

2 occurs a maximum of three times. So, 2 X 2 X 2 = 8 3 occurs a maximum of one time. So, 3 X 1 = 3. 5 occurs a maximum of one time. So, 5 X 1 = 5. Therefore, LCM = 8 X 3 X 5 = 120. |

**Example 2: Find the HCF and LCM of \(\frac{4}{9}\) and \(\frac{6}{21}\).**

**Example 2: Find the HCF and LCM of \(\frac{4}{9}\) and \(\frac{6}{21}\).**

Numerators of the two fractions: 4 and 6

Prime factors of 4 and 6:

4 = 2 X 2 6 = 2 X 3 |

HCF of 4 and 6 is 2.

LCM of 4 and 6 is: 2 X 2 X 3 = 12

Denominators of the two fractions: 9 and 21

Prime factors of 9 and 21:

9 = 3 X 3 21 = 3 X 7 |

HCF of 9 and 21 is 3.

LCM of 9 and 21: 3 X 3 X 7 = 63.

f. LCM of \(\frac{4}{9}\) and \(\frac{6}{21}\) = \(\frac{LCM of Numerators}{HCF of Denominators}\) = \(\frac{12}{3}\) = 4

g. HCF of \(\frac{4}{9}\) and \(\frac{6}{21}\) = \(\frac{HCF of Numerators}{LCM of Denominators}\) = \(\frac{2}{63}\)

So, now you know the definition, formulas, and properties of HCF and LCM. You have also gone through some examples elaborating on these concepts. Solve more questions form the textbooks and master them.

### HCF and LCM – Sample Questions

Q1: Two numbers are in the ratio of 5:11. If their HCF is 7, find the numbers.Ans: Let the numbers be 5m and 11m. Since 5:11 is already the reduced ratio, ‘m’ has to be the HCF. So, the numbers are 5 x 7 = 35 and 11 x 7 = 77. |

Q2: Find the greatest number which on dividing 70 and 50 leaves remainders 1 and 4 respectively. Ans: The required number leaves remainders 1 and 4 on dividing 70 and 50 respectively. This means that the number exactly divides 69 and 46.So, we need to find the HCF of 69 (3 x 23) and 46 (2 x 23). HCF (69, 46) = 23 Thus, 23 is the required number. |

Q3: Find the least number which when divided by 5,7,9 and 12, leaves the same remainder 3 in each case.Ans: In these type of questions, we need to find the LCM of the divisors and add the common remainder (3) to it.So, LCM (5, 7, 9, 12) = 1260 Therefore, required number = 1260 + 3 = 1263. |

Q4: Find the least number which when divided by 6,7,8 leaves a remainder 3, but when divided by 9 leaves no remainder. Ans: LCM (6, 7, 8) = 168So, the number is of the form 168m + 3. Now, 168m + 3 should be divisible by 9. We know that a number is divisible by 9 if the sum of its digits is a multiple of 9. For m = 1, the number is 168 + 3 = 171, the sum of whose digits is 9. Therefore, the required number is 171. |

Q5: A rectangular field of dimension 180m x 105m is to be paved by identical square tiles. Find the size of each tile and the number of tiles required. Ans: We need to find the size of a square tile such that a number of tiles cover the field exactly, leaving no area unpaved.For this, we find the HCF of the length and breadth of the field. HCF (180, 105) = 15 Therefore, size of each tile = 15m x 15m Also, number of tiles = area of field / area of each tile => Number of tiles = (180 x 105) / (15 x 15) => Number of tiles = 84 Hence, we need 84 tiles, each of size 15m x 15m. |

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