Most prime numbers end in 1, 3, 7, or 9 because any whole number ending in 0, 2, 4, 5, 6, or 8 is automatically divisible by 2 or 5. The only prime exceptions are 2 and 5 themselves.
Direct Answer
Every prime number greater than 5 must end in 1, 3, 7, or 9 when written in base 10. These last digits are not a guarantee of primality. They only show that the number is not immediately divisible by 2 or 5.
Why Last Digits Matter for Prime Numbers
A prime number is a whole number greater than 1 with exactly two positive divisors: 1 and itself. If a number has another positive divisor, it is composite.
In the decimal system, the last digit tells whether a number is divisible by 2 or by 5. Since every even number greater than 2 is composite, and every number ending in 5 greater than 5 is divisible by 5, many last digits are ruled out right away.
Decimal Last-Digit Rule
For primes greater than 5, the possible last digits are exactly 1, 3, 7, and 9. These are the decimal digits that are not divisible by 2 or 5.
Which Ending Digits Are Ruled Out?
The table below shows why six decimal endings cannot belong to a prime number greater than 5.
| Last Digit | Can a Prime Greater Than 5 End This Way? | Reason |
|---|---|---|
| 0 | No | Divisible by 10, so also divisible by 2 and 5. |
| 1 | Yes | Not automatically divisible by 2 or 5. |
| 2 | No | Even numbers greater than 2 are composite. |
| 3 | Yes | Not automatically divisible by 2 or 5. |
| 4 | No | Even numbers greater than 2 are composite. |
| 5 | No | Numbers ending in 5 are divisible by 5. |
| 6 | No | Even numbers greater than 2 are composite. |
| 7 | Yes | Not automatically divisible by 2 or 5. |
| 8 | No | Even numbers greater than 2 are composite. |
| 9 | Yes | Not automatically divisible by 2 or 5. |
The Four Possible Endings Are Prime Candidates
A number ending in 1, 3, 7, or 9 is only a prime candidate. It has passed a quick divisibility filter, but it still may have other factors.
Can Be Prime
11, 13, 17, 19, 31, 37, and 41 all end in 1, 3, 7, or 9 and are prime.
Can Be Composite
21, 33, 49, 77, 91, and 99 also end in those digits, but each has a divisor other than 1 and itself.
Common Mistake
Ending in 1, 3, 7, or 9 does not prove that a number is prime. For example, 91 ends in 1, but 91 = 7 × 13, so it is composite.
How This Connects to Modular Arithmetic
The last digit of a decimal number is its remainder when divided by 10. A number ending in 7, for example, has the same remainder as 7 after division by 10.
Prime Ending Rule in Mod 10
If p is a prime greater than 5, then p mod 10 is 1, 3, 7, or 9.
This works because 10 = 2 × 5. Any number with a remainder of 0, 2, 4, 5, 6, or 8 modulo 10 has a direct connection to divisibility by 2 or 5. Those cases cannot produce primes greater than 5.
Why 2 and 5 Are Exceptions
The primes 2 and 5 are special because they are the divisors used in the decimal base itself. The number 2 is the only even prime, and 5 is prime because its only positive divisors are 1 and 5.
After 5, any number ending in 2 is an even number greater than 2, and any number ending in 5 has 5 as a smaller positive divisor. That is why the pattern starts after these two one-digit primes.
Pattern Note
The rule depends on writing numbers in base 10. It is a decimal ending pattern, not a claim that prime numbers follow a simple repeating formula.
Ending Digits Narrow the Search, Not the Proof
Primality testing needs more than a last-digit check. Trial division, the square root test, and sieving methods look for possible divisors beyond 2 and 5.
For a number n, the square root test works because every composite number has a factor pair. If one factor is greater than √n, the matching factor must be less than √n. This is why testing possible divisors up to √n is enough for a basic primality check.
Check a Number Directly
To test a specific value, use PrimeOrNot’s prime number checker and compare the result with the last-digit rule explained here.
Test a Prime Candidate
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Related Concepts
FAQ
Do All Primes End in 1, 3, 7, or 9?
No. The primes 2 and 5 do not end in 1, 3, 7, or 9. Every prime greater than 5 does.
Is Every Number Ending in 1, 3, 7, or 9 Prime?
No. These endings only remove numbers that are divisible by 2 or 5. A number ending in 1, 3, 7, or 9 can still be composite, such as 21, 49, 77, or 91.
Why Can’t a Prime Greater Than 5 End in 5?
Any whole number greater than 5 that ends in 5 is divisible by 5. Since it has 5 as a smaller positive divisor, it cannot be prime.
Why Can’t a Prime Greater Than 2 Be Even?
Every even number greater than 2 is divisible by 2. That gives it a divisor other than 1 and itself, so it is composite.
Does the Last Digit Pattern Predict Prime Numbers?
No. It only filters out clear composite cases. A full primality test still needs to check whether the number has other divisors.