No — 1 is not a prime number, and it is not composite either. A prime needs exactly two divisors, 1 and itself, but 1 has only one divisor: itself. A composite needs more than two divisors, and 1 has just one. In number theory 1 is called a unit — the only number that is neither prime nor composite.
1 is the most argued-about number in elementary mathematics. It looks prime — it cannot be broken into smaller whole-number factors, and it divides everything — yet every textbook says it is not prime. The reason is not a technicality. It is one sentence long: a prime number has exactly two divisors, and 1 has only one. This post proves it with the divisor table, explains what 1 actually is, and shows the deeper reason the definition is written this way.
Check the verdict yourself with a free prime number checker — enter 1 and it returns “neither”, enter 2 and it returns “prime”.
The Definition Test: Count the Divisors
A prime number is a whole number greater than 1 with exactly two divisors: 1 and itself. A composite number is a whole number greater than 1 with more than two divisors. Both definitions start with “greater than 1” — and that is where 1 fails before the counting even starts. But the counting is the clearer proof:
| Number | Divisors | How many? | Verdict |
|---|---|---|---|
| 1 | 1 | 1 | Neither prime nor composite |
| 2 | 1, 2 | 2 | Prime — smallest prime |
| 3 | 1, 3 | 2 | Prime |
| 4 | 1, 2, 4 | 3 | Composite |
| 5 | 1, 5 | 2 | Prime |
| 6 | 1, 2, 3, 6 | 4 | Composite |
| 7 | 1, 7 | 2 | Prime |
| 9 | 1, 3, 9 | 3 | Composite |
2 has two divisors, so it is prime. 4 has three, so it is composite. 1 has one — and one is neither two nor more than two, so 1 fits neither definition. The test is a count, and 1 simply does not reach the minimum.
The same table answers the follow-up question students always ask next: is 0 a prime number? No — and 0 is not composite either. 0 is divisible by every whole number except itself, so it has infinitely many divisors, which fits neither “exactly two” nor any sensible count. 0 and 1 are the only two whole numbers that sit outside both categories.
Why 1 Is Not Composite Either
It is easy to see why 1 is not prime — it has too few divisors. It is less obvious why it is not composite, because “composite” sounds like it should mean “not prime”. It does not. Composite means something specific: a whole number greater than 1 that can be written as a product of smaller whole numbers greater than 1. 1 cannot be written that way — the only product that gives 1 is 1 × 1, and 1 is not greater than 1. So 1 fails the composite test for the same reason it fails the prime test: it does not have enough divisors, and it is not greater than 1.
In number theory, 1 has its own name: a unit. A unit is a number that divides every other number — and 1 divides everything, because any number divided by 1 is itself. Units are the third category, alongside primes and composites, and for the whole numbers the units are just 1 (and −1, if negative numbers are allowed). Saying “1 is a unit” is not a dodge; it is the precise classification, the same way “0 is neither positive nor negative” is precise.
The Deeper Reason: Unique Factorization
The one-divisor argument is the proof, but it is not the reason mathematicians chose the definition. The reason is the Fundamental Theorem of Arithmetic: every whole number greater than 1 can be written as a product of primes in exactly one way (apart from order). 6 = 2 × 3, and that is the only way. 91 = 7 × 13, and that is the only way. That uniqueness is the foundation of almost everything in number theory — it is what makes prime factorization a reliable tool instead of a guessing game.
If 1 were prime, uniqueness would collapse. 6 = 2 × 3, but also 6 = 1 × 2 × 3, and 6 = 1 × 1 × 2 × 3, and 6 = 1 × 1 × 1 × 2 × 3 — infinitely many factorizations, all different, all “correct”. Every number would have infinitely many prime factorizations, and the theorem would be false. Excluding 1 from the primes is what keeps factorization unique: with 1 out, each number has exactly one prime factorization, and the whole system holds together.
That is also why the definition says “greater than 1” instead of just “exactly two divisors”. The two wordings agree on every number except 1, and the “greater than 1” version makes the exclusion explicit — 1 is out by definition, so the theorem survives.
A Short History: 1 Used to Be Called Prime
The confusion has a history. Some older textbooks did call 1 prime, and the convention only settled in the early 20th century, when mathematicians agreed that unique factorization mattered more than the old habit. That is why the mistake survives: a grandparent’s textbook, an old puzzle book, or a half-remembered lesson can still say 1 is prime, and the student who trusts it is not being careless — they are repeating a convention that was once standard.
The modern rule is unambiguous, and every current source agrees: 1 is neither prime nor composite. The primes start at 2 — the smallest prime, and the only even prime — and 1 sits alone in its own category, the unit.
The Smallest Prime Is 2, Not 1
If 1 is out, the primes begin at 2, and 2 is a strange prime in its own right: it is the only even prime, because every other even number is divisible by 2 as well as by 1 and itself. The primes under 20 are 2, 3, 5, 7, 11, 13, 17 and 19 — eight primes, starting at 2, with 1 nowhere in the list. There are 25 primes from 1 to 100, and the count starts at 2 as well: the range is called “1 to 100” only because 1 is where counting starts, not because 1 is prime.
That is the habit to take away: when a question asks “is this number prime?”, the first step is not dividing — it is checking the number is greater than 1. If it is 1 or 0, the answer is “neither” before any division happens.
Frequently Asked Questions
Is 1 a prime number?
No — 1 is neither prime nor composite. A prime number has exactly two divisors, 1 and itself, but 1 has only one divisor: itself. A composite number needs more than two divisors, and 1 has just one.
Why is 1 not a prime number?
Because it has only one divisor, and a prime needs exactly two. The definition also requires the number to be greater than 1, which 1 fails directly. The deeper reason is unique factorization: if 1 were prime, 6 = 2 × 3 would also equal 1 × 2 × 3 and 1 × 1 × 2 × 3, and factorization would no longer be unique.
Is 1 prime or composite?
Neither. Prime means exactly two divisors; composite means more than two divisors (and greater than 1). 1 has one divisor, so it fits neither category. In number theory it is called a unit.
Is 0 a prime number?
No — 0 is neither prime nor composite either. 0 is divisible by every whole number, so it has infinitely many divisors. Only whole numbers greater than 1 are sorted into prime and composite.
What is the smallest prime number?
2. It is the smallest prime and the only even prime — its divisors are 1 and 2, exactly two. The primes under 20 are 2, 3, 5, 7, 11, 13, 17 and 19.
Is 1 a prime number in old textbooks?
Some older books did call 1 prime, but the modern convention excludes it. The change settled in the early 20th century to protect unique prime factorization. Every current source agrees: 1 is neither prime nor composite.
Try It Yourself
Enter 1 into a free prime number checker and see the “neither” verdict — then enter 0, 2 and 4 and watch the three categories line up. For the number that fools people in the other direction, read is 91 prime — 91 = 7 × 13, so it has four divisors and is composite, not prime. And for the full factorization of any number, the Factor Calculator breaks it into primes and gives the GCF and LCM too. To go deeper into the definition itself, see is 1 a prime number on the calculator page.