What would be HCF of 1000 consecutive prime numbers
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Co-prime Numbers
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Co-prime numbers are the numbers whose common factor is only 1. There should be a minimum of two numbers to form a set of co-prime numbers. Such numbers have only 1 as their highest common factor, for example, {4 and 7}, {5, 7, 9} are co-prime numbers. Co-prime numbers need not be prime numbers always. Two composite numbers like 4 and 9 also form a pair of co-primes.
Answer:
1
Step-by-step explanation:
HCF or Highest Common Factor is the greatest number that divides each of the two or more numbers.
HCF (of Any Primes) is 1. So they are coprime. So naturally, the HCF of 1000 Primes is 1.
The only factor of a prime number (other than itself) is 1. A prime number cannot have another prime factor. So, obviously, the Highest Common Factor of any two prime numbers (not just consecutive) is 1, since it is the only common factor.
The Euler's Theorem [a^(phi-1)=1 (mod m) ] to work, the first requisite is that (m, a) should be coprime, or else there may not be a Multiplicative Inverse and the Theorem fails to work. This is the reason why we always search for a Larger Prime Number in Cryptography as Modulo. Also, this is the only reason why our Scientists /Mathematicians are trying to find out larger and larger Primes for cryptography. So while fixing the value of phi (Euler's totient number ), we choose all numbers which are co-primes to (m) and count such numbers to fix the value of phi in Formula /Formulae.
The HCF of any 1000 Primes is 1.