**Prime Number:** A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
For example, 2, 3, 5, 7, 11, 13, etc. are prime numbers.

**Co-Prime Number:** Two numbers are said to be relatively prime, mutually prime, or co-prime to each other, when they have no common factor or the only common positive factor of the two numbers, is 1.
In other words, two numbers are said to be Co-prime if their H.C.F. is 1.

**Factors:** The numbers are said to be factors of a given number when they exactly divide that number.
Thus, factors of 18 are 1, 2, 3, 6, 9 and 18.

**Common Factors:** A common factor of two or more numbers is a number which divides each of them exactly.
Thus, each of the numbers - 2, 4 and 8 is a common factor of 8 and 24.

**Multiple:** When a number is exactly divisible by another number, then the former number is called the multiple of the latter number.
Thus, 45 is a multiple of 1, 3, 5, 9, 15 and 45.

**Common Multiple:** A common multiple of two or more numbers is a number which is exactly divisible by each of them.

For example, 12, 24 and 36 is a common multiple of 3, 4, 6 and 12.

**Prime Factorisation:** If a natural number is expressed as the product of prime numbers, then the factorization of the number is called its prime factorization. Prime factorization of a natural number can be expressed in the exponential form.

For example: • 24 = 2 x 2 x 2 x 3 = 2² x 3 • 420 = 2 x 2 x 3 x 5 x 7 = 2² x 3 x 5 x 7

**Highest Common Factor (H.C.F.) or Greatest Common Divisor (G.C.D.) or Greatest Common Measure (G.C.M.) are synonymous terms:**

The H.C.F of two or more than two numbers is the greatest numbers which divide each of them without any remainder.

Methods of finding the H.C.F. of a given set of numbers:

**Method I: Prime Factorisation method :**
Express each one of the given numbers as the product of prime factors. The product of least powers/index of common prime factors gives H.C.F.

**Example I:**

**Q.Find the H.C.F. of 8 and 14 by Prime Factorisation method?**
**Solution:**
8 = 2 x 2 x 2
14 = 2 x 7
Common factor of 8 and 14 = 2.
Thus, the Highest Common Factor (H.C.F.) of 8 and 14 = 2.

**Example II:**

**Q.Find the H.C.F. of 24, 36 and 72 by Prime Factorisation method?**
**Solution:**
24 = 2 x 2 x 2 x 3
36 = 2 x 2 x 3 x 3
72 = 2 x 2 x 2 x 3 x 3
H.C.F. of 24, 36 and 72 = Product of common factors with least powers/index = 2^2 x 3
Thus, Highest Common Factor (H.C.F.) of 24, 36 and 72 = 12

**Method II: Successive Division method :**

Divide the larger number by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is the required H.C.F.

**Example I:**

**Q.Find the H.C.F. of 8 and 14 by Successive Division method? **
**Solution:**
8 | 14 | 1
8
6 | 8 | 1
6
2 | 6 | 3
6
0

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**LCM & HCF Quiz**

**Q1.The product of two 2 digit numbers is 2028 and their HCF is 13. What are the numbers?**

**Sol:**
Let the numbers be 13x and 13y (? HCF of the numbers = 13)
13x × 13y = 2028
=> xy = 12
co-primes with product 12 are (1, 12) and (3, 4) (? we need to take only co-primes with product 12. If we take two numbers with product 12, but not co-prime, the HCF will not remain as 13)
Hence the numbers with HCF 13 and product 2028
= (13 × 1, 13 × 12) and (13 × 3, 13 × 4)
= (13, 156) and (39, 52)
Given that the numbers are 2 digit numbers
Hence numbers are 39 and 52

**Q2.Six bells commence tolling together and toll at intervals of 2, 4, 6, 8 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together?**

**Sol:**
L.C.M. of 2, 4, 6, 8, 10, 12 is 120.
So, the bells will toll together after every 120 seconds(2 minutes).
In 30 minutes, they will toll together 30/2 + 1 = 16 times.

**Q3.N is the greatest number which divides 1305, 4665 and 6905 and gives the same remainder in each case. What is the sum of the digits in N?**

**Sol:**
If the remainder is the same in each case and the remainder is not given, HCF of the differences of the numbers is the required greatest number
6905 - 1305 = 5600
6905 - 4665 = 2240
4665 - 1305 = 3360
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