All 25 primes under 100, how to find them yourself, and a two-second test for any number
There are 25 prime numbers between 1 and 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. A prime is a whole number greater than 1 whose only factors are 1 and itself. 1 is not prime, and 2 is the only even prime.
The count thins out as the numbers grow: four primes in the first ten, only one in the nineties. Every prime above 5 ends in 1, 3, 7 or 9, because anything ending in an even digit is divisible by 2 and anything ending in 5 or 0 is divisible by 5.
| Range | Primes | Count |
|---|---|---|
| 1–10 | 2, 3, 5, 7 | 4 |
| 11–20 | 11, 13, 17, 19 | 4 |
| 21–30 | 23, 29 | 2 |
| 31–40 | 31, 37 | 2 |
| 41–50 | 41, 43, 47 | 3 |
| 51–60 | 53, 59 | 2 |
| 61–70 | 61, 67 | 2 |
| 71–80 | 71, 73, 79 | 3 |
| 81–90 | 83, 89 | 2 |
| 91–100 | 97 | 1 |
Write out 2 to 100. Circle 2, then cross out every other multiple of 2 (4, 6, 8 …). Circle the next number still standing, 3, and cross out its multiples (9, 15, 21 …). Do the same for 5 and for 7.
Then stop. Every composite number up to 100 has a factor no bigger than 10, because 11 × 11 = 121 is already past 100. So once 2, 3, 5 and 7 have been sieved, everything left standing is prime — exactly the 25 numbers above.
Only try dividing by the primes up to the square root of the number. For anything under 100 that means 2, 3, 5 and 7 — four checks. For a number under 400, add 11, 13, 17 and 19.
The divisibility rules make those checks instant: an even last digit means 2 divides it, a digit sum that is a multiple of 3 means 3 does, a last digit of 0 or 5 means 5 does. Seven is the only one that usually needs actual division.
The numbers people most often mistake for primes are odd numbers that just look awkward: 51 = 3 × 17, 57 = 3 × 19, 87 = 3 × 29, 91 = 7 × 13. Each fails one of the four checks.
Every whole number above 1 is either prime or a product of primes in exactly one way — 60 = 2 × 2 × 3 × 5 and nothing else. That is why primes are called the building blocks of arithmetic, and why simplifying a fraction or finding a common denominator is really a question about shared prime factors.
Primes also run modern encryption. Multiplying two huge primes is easy; working backwards from the product to the two primes is so slow that it keeps online banking secure.
Every number is a product of the primes above. These are the first few written out — primes stand alone.
| Number | Prime factors | Prime? |
|---|---|---|
| 2 | 2 | Yes |
| 3 | 3 | Yes |
| 4 | 2 × 2 | No |
| 5 | 5 | Yes |
| 6 | 2 × 3 | No |
| 7 | 7 | Yes |
| 8 | 2 × 2 × 2 | No |
| 9 | 3 × 3 | No |
| 10 | 2 × 5 | No |
| 11 | 11 | Yes |
| 12 | 2 × 2 × 3 | No |
| 13 | 13 | Yes |
| 14 | 2 × 7 | No |
| 15 | 3 × 5 | No |
| 16 | 2 × 2 × 2 × 2 | No |
| 17 | 17 | Yes |
| 18 | 2 × 3 × 3 | No |
| 19 | 19 | Yes |
| 20 | 2 × 2 × 5 | No |
| 21 | 3 × 7 | No |
| 22 | 2 × 11 | No |
| 23 | 23 | Yes |
| 24 | 2 × 2 × 2 × 3 | No |
| 25 | 5 × 5 | No |
| 26 | 2 × 13 | No |
| 27 | 3 × 3 × 3 | No |
| 28 | 2 × 2 × 7 | No |
| 29 | 29 | Yes |
| 30 | 2 × 3 × 5 | No |
25. They are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
No. A prime has exactly two different factors, 1 and itself. The number 1 has only one factor, so it is neither prime nor composite.
Yes. Its only factors are 1 and 2, which makes it the smallest prime and the only even one — every other even number is divisible by 2.
No. 91 = 7 × 13. It is the most commonly mistaken number under 100 because it is odd, does not end in 5 and its digits add to 10.
97. The next prime after it is 101.
Primes are the backbone of simplifying fractions and of the divisibility rules page; the division calculator shows the remainder whenever a number does not divide exactly.