What Makes a Number Prime? Understanding Primes for Students
1 is a very unique number. In classical theory it is neither considered even, odd or anything. It is just considered as a building block for any other number. 1 is so unique that it can not be included in any group nor in the group of prime numbers. It has been seen in the ancient time there was some ambiguity regarding inclusion of 1 in the set of prime numbers. Even the great mathematician G.H.Hardy seems to be in little confusion as he included 1 in the set of prime numbers in the first six editions of his book ``A Course in Pure Mathematics” till 1933. But in 1938 he updated the inclusion and considered 2 to be the first prime number to start with.
Prime Number
Before knowing if 1 is a prime number or not, let us understand what is a prime number. A prime number may be an integer greater than 1 whose only factors are 1 and itself. An element may be an integer which will be divided evenly into another number. The few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and so on. Numbers that have more than two factors are called composite numbers. The number 1 is neither prime nor composite.
For every prime p, there exists a major number p' such p' is bigger than p. This proof, which was demonstrated in the past by the Greek mathematician Euclid, validates the concept that there's no "largest" prime. As the set of natural numbers N = {1, 2, 3, ...} proceeds, prime numbers are generally subsided frequently and are harder to seek out in a reasonable amount of time. As of this writing, the most important known prime has 24,862,048 digits. It was discovered in 2018 by Patrick Laroche of the good Internet Mersenne Prime Search (GIMPS).
Properties of Prime Numbers
Every number that's greater than 1 is often divided by a minimum of one prime.
Every even positive integer greater than the amount 2 is often expressed because of the sum of two primes.
List of Prime Numbers
Is 1 a Prime Number?
Number 1 has positive divisors for 1 and itself. consistent with the definition of a prime number. Any number having only two positive divisors is referred to as prime numbers. So, is 1 a prime number or not? Is 1 a prime or composite number?
The answer to the present question is: No, 1 isn't a major prime number and it's not a composite number!
Lesson Summary:
Why is 1 not a Prime Number?
The answer to Why 1 is not a Prime Number is present in the definition for the prime numbers itself. For a number to be called the prime number, it must have only two of the positive factors. Now, for 1, the number of positive divisors or factors is only one that is 1 itself. So, this is why 1 is not a prime number here. But it is the most important number in Mathematics as it is the basic number used for forming other numbers.
Note: 2 is the smallest number that satisfies the definition for the prime numbers.
Solved Examples
Question 1: Which one of the following is a prime number?
13
6
10
4
Answer: 13 is a prime number because 13 has only two factors that are 13 and 1.
Question 2: Which one of the following is a prime number?
3
16
20
15
Answer: 3 is a prime number because 3 has only two factors that are 3 and 1.
Question 3: Which one of the following is a prime number?
12
43
90
15
Answer: 43 is a prime number because 43 has only two factors that are 43 and 1.
Question 4: Which one of the following is a prime number?
21
63
53
15
Answer: 53 is a prime number because 53 has only two factors that are 53 and 1.
FAQs on Is 1 a Prime Number?
1. What is the official definition of a prime number according to the NCERT syllabus?
According to the NCERT syllabus, a prime number is a natural number greater than 1 that has exactly two distinct positive factors: 1 and the number itself. For a number to be classified as prime, it must meet both of these conditions. For instance, the number 5 is prime because its only factors are 1 and 5.
2. Why isn't the number 1 considered a prime number?
The number 1 is not a prime number because it fails to meet a critical part of the definition: it does not have two distinct factors. A prime number must be divisible by 1 and itself, and these two factors must be different. For the number 1, the factors are 1 and 1, which are the same. Since it has only one unique factor, it is not a prime number.
3. Is 1 a prime number or a composite number?
The number 1 is classified as neither prime nor composite. Here’s a simple explanation:
Not Prime: It has only one factor (itself), not the required two distinct factors.
Not Composite: A composite number is a natural number with more than two factors. Since 1 only has one factor, it cannot be composite.
Therefore, 1 exists in its own special category.
4. If 1 is not prime or composite, what type of number is it?
While 1 is not prime or composite, it is classified in several other ways in mathematics. The number 1 is a:
Natural Number: A positive integer used for counting (1, 2, 3, ...).
Odd Number: An integer that is not divisible by 2.
Rational Number: It can be expressed as a fraction, such as 1/1.
Multiplicative Identity: Any number multiplied by 1 remains the same.
5. What is the core difference between the factors of 1 and the factors of a prime number like 3 or 7?
The core difference is the uniqueness of the factors. For any prime number, such as 7, its factors are 1 and 7—two different and distinct numbers. In contrast, for the number 1, its only factor is 1. It lacks a second, different factor, which is the fundamental reason it is excluded from the set of prime numbers.
6. Why is it so important for mathematics that 1 is not treated as a prime number?
Excluding 1 from the list of primes is essential for upholding the Fundamental Theorem of Arithmetic. This theorem states that every integer greater than 1 can be written as a unique product of prime numbers. For example, the number 30 is uniquely factored as 2 × 3 × 5. If 1 were considered prime, you could create infinite factorizations (e.g., 1 × 2 × 3 × 5, 1 × 1 × 2 × 3 × 5), and the concept of a single, unique prime factorization would be destroyed.
7. When did mathematicians officially decide that 1 is not a prime number?
The modern consensus that 1 is not a prime number became firmly established in the late 19th and early 20th centuries. While earlier mathematicians sometimes had differing views, the need to preserve the uniqueness of the Fundamental Theorem of Arithmetic led to the widespread adoption of the definition that strictly excludes 1.
8. Why is 2 the only even prime number, and how does this relate to the number 1?
The number 2 is the only even prime because its only factors are 1 and 2 (two distinct factors). Every other even number (4, 6, 8, etc.) is divisible by 2, meaning it will always have at least three factors: 1, 2, and itself. This makes all other even numbers composite. This concept highlights the strictness of the 'exactly two distinct factors' rule, which is the same rule that disqualifies the number 1 from being prime.
