No, the smallest prime number and the smallest composite number are not coprime.
Understanding Prime and Composite Numbers
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The list of prime numbers begins with 2, 3, 5, 7, and so on. The smallest prime number is 2. It is also the only even prime number.
A composite number is a natural number greater than 1 that is not prime. It can be formed by multiplying two smaller positive integers. The list of composite numbers begins with 4, 6, 8, 9, 10, and so on. The number 1 is considered neither prime nor composite. The smallest composite number is 4.
The Concept of Coprime Numbers
Two integers are said to be coprime (or relatively prime) if their only common positive divisor is 1. In other words, their greatest common divisor (GCD) is 1.
For example, the numbers 7 and 10 are coprime. The divisors of 7 are 1 and 7. The divisors of 10 are 1, 2, 5, and 10. The only common divisor is 1, so their GCD is 1.
Determining if 2 and 4 are Coprime
To determine if the smallest prime number (2) and the smallest composite number (4) are coprime, we need to find their greatest common divisor.
- The divisors of 2 are 1 and 2.
- The divisors of 4 are 1, 2, and 4.
The common divisors of 2 and 4 are 1 and 2. The greatest among these is 2.
Since the greatest common divisor of 2 and 4 is 2, and not 1, the two numbers are not coprime.
Justification and Generalization
The justification for this result lies in the fundamental definitions of the numbers themselves. The smallest prime number, 2, is a factor of the smallest composite number, 4 (4=2×24 equals 2 cross 2
4=2×2
). Whenever one number is a factor of another number, their greatest common divisor is the smaller of the two numbers. In this case, since 2 is a factor of 4, the GCD of 2 and 4 is 2.
This relationship holds true more generally: if a prime number pp
𝑝
is a factor of another number nn
𝑛
, then pp
𝑝
and nn
𝑛
cannot be coprime, because their greatest common divisor will be at least pp
𝑝
(and therefore not 1).
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