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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