# If p and q are two prime number, then what is their HCF?

By BYJU'S Exam Prep

Updated on: September 25th, 2023

The HCF of two prime numbers p and q is 1. Prime numbers are natural numbers greater than 1 that can only be divided by 1 and themselves without leaving a remainder. In other words, they have exactly two distinct positive divisors: 1 and the number itself.

For example- The first few prime numbers are 2, 3, 5, 7, 11, and so on. These numbers are not divisible by any other number except 1 and themselves.

Table of content

## Prime Numbers 1 to 100

Check out the detailed overview of prime numbers between 1 and 100 to have a better understanding of prime numbers.

 Prime Numbers 1 to 100 Prime Numbers 1 to 10 2, 3, 5, 7 Prime Numbers 10 to 20 11, 13, 17, 19 Prime Numbers 20 to 30 23, 29 Prime Numbers 30 to 40 31, 37 Prime Numbers 40 to 50 41, 43, 47 Prime Numbers 50 to 60 53, 59 Prime Numbers 60 to 70 61, 67 Prime Numbers 70 to 80 71, 73, 79 Prime Numbers 80 to 90 83, 89 Prime Numbers 90 to 100 97

## Key Features Of Prime Numbers

Prime numbers play a significant role in number theory and have practical applications in various fields such as cryptography, computer science, and mathematics. Have a look at key features of prime numbers here-

• The first prime number is 2, which is the only even prime number. All other even numbers greater than 2 are divisible by 2 and therefore not prime.
• The prime numbers between 1 and 100 include: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
• The sum of all the prime numbers between 1 and 100 is 1,241.
• The largest two-digit prime number is 97.
• The smallest two-digit prime number is 11.
• There are a total of 25 prime numbers between 1 and 100.
• Prime numbers have various applications in cryptography, number theory, and computer science.
• The prime number theorem, proved by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896, gives an approximation for the number of primes less than a given value.
• Prime numbers are often used in generating secure encryption keys.
• Prime numbers play a crucial role in many algorithms, such as the Sieve of Eratosthenes, which is used to find prime numbers efficiently.

Solution:

To prove that the highest common factor (HCF) of two prime numbers, p and q, is 1.

Assume that p and q are prime numbers

The prime factorization of p is p = p * 1,

and the prime factorization of q is q = q * 1.

Now, let’s find the common factors of p and q.

(A common factor is a number that divides both p and q without leaving any remainder.)

If d is a common factor of p and q,

It means that d divides both p and q completely.

In terms of prime factorization, this means that d must be a common prime factor of p and q.

However, since p and q are both prime numbers,

their only prime factors are themselves.

Therefore, the only common prime factor of p and q is 1.

Hence, the highest common factor (HCF) of two prime numbers p and q is 1, as there are no prime factors other than 1 that are common to both p and q.

Summary:

## If p and q are two prime number, then what is their HCF?

If p and q are two prime numbers, then their HCF is 1. To find the HCF; one must use the technique of Prime Factorization. The prime factorization of a number represents the unique combination of prime factors that, when multiplied together, give the original number.

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