Prime Numbers - Prime Numbers 1 to 100 With Examples

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Prime Numbers - Prime Numbers 1 to 100 With Examples

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Bhanzu Team Last updated on May 10, 20268 min read

A prime number is an integer greater than 111 that has exactly two positive divisors — itself and 111. Every other integer greater than 111 is composite — it can be written as a product of two smaller integers. The first ten primes are 2,3,5,7,11,13,17,19,23,292, 3, 5, 7, 11, 13, 17, 19, 23, 292,3,5,7,11,13,17,19,23,29, and there are infinitely many. Primes are the indivisible building blocks of every other integer, and they sit at the heart of modern cryptography.

Quick Reference

Field Value
Definition An integer >1> 1>1 with exactly two positive divisors: 111 and itself
Symbol No standard symbol; the set is sometimes denoted P\mathbb{P}P
First 10 primes 2,3,5,7,11,13,17,19,23,292, 3, 5, 7, 11, 13, 17, 19, 23, 292,3,5,7,11,13,17,19,23,29
Number of primes Infinitely many (proved by Euclid, c. 300 BCE)
Used in Number theory, RSA cryptography, hashing, error-correcting codes

What is a Prime Number?

A prime number has exactly two divisors: 111 and itself. The number 777 is prime because the only positive integers that divide it cleanly are 111 and 777. The number 999 is not prime because 333 also divides it — so 999 has three divisors (1,3,91, 3, 91,3,9).

The number 111 is not prime, by convention. If 111 were allowed, the unique prime factorisation of every other integer would break — so mathematicians defined 111 out of the set. The number 222 is the only even prime; every other even number is divisible by 222, so it has at least three divisors and cannot be prime.

Prime Numbers List — 1 to 100

There are exactly 252525 primes from 111 to 100100100:

2,3,5,7,11,13,17,19,23,29,2, 3, 5, 7, 11, 13, 17, 19, 23, 29,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,31, 37, 41, 43, 47, 53, 59, 61, 67, 71,31,37,41,43,47,53,59,61,67,71,73,79,83,89,9773, 79, 83, 89, 9773,79,83,89,97

The primes thin out as numbers grow. There are 252525 primes below 100100100, but only 212121 between 100100100 and 200200200, and only 161616 between 200200200 and 300300300. The thinning is captured by the Prime Number Theorem: the count of primes up to nnn is approximately nln⁡n\frac{n}{\ln n}lnnn​.

How to Identify a Prime — The Sieve of Eratosthenes

The cleanest method to find all primes up to a limit NNN is the sieve developed by Eratosthenes of Cyrene around 200200200 BCE:

  1. List all integers from 222 to NNN.

  2. Circle 222 (the first prime). Cross out every multiple of 222 that is greater than 222.

  3. Move to the next un-crossed number (333). Circle it. Cross out every multiple of 333 greater than 333.

  4. Continue with the next un-crossed number, repeating.

  5. Stop when you reach N\sqrt{N}N​. Every remaining un-crossed number is prime.

The sieve gives the primes 111 to 100100100 in a few minutes by hand, and it is the foundation of every modern prime-testing algorithm.

Why Prime Numbers Matter

The primes are the building blocks of arithmetic. Euclid's Fundamental Theorem of Arithmetic states that every integer greater than 111 can be written as a product of primes in exactly one way (up to ordering). The number 60=2×2×3×560 = 2 \times 2 \times 3 \times 560=2×2×3×5, and no other product of primes equals 606060. This unique factorisation is what makes primes the atoms of integer arithmetic.

In modern cryptography, primes do the heavy lifting. RSA encryption — the system that protects bank transactions, email, and most internet communication — relies on the fact that multiplying two large primes is easy, but factoring the product back into those primes is computationally hard. Two primes with 300300300 digits each multiply in a fraction of a second; their product would take more than the age of the universe to factor on the fastest known computer.

Cryptography is where prime numbers stopped being a curiosity of pure math and became infrastructure.

[MATHEMATICIANS & HISTORY CALLOUT]

Title: Euclid's proof that primes never run out

Mathematician: Euclid of Alexandria (c. 300 BCE, Greece)

Date and place: Alexandria, c. 300 BCE

The story: In Elements Book IX, Proposition 20, Euclid proved that there are infinitely many primes — using a single short argument that has stood for 2,3002{,}3002,300 years. He assumed there were only finitely many primes, multiplied them all together, added 111, and showed that the result either was a new prime or had a prime factor not in the original list. Either way, the assumption was wrong; there must be more primes than any finite list. The proof is one of the cleanest in mathematics, requires no machinery, and is still taught in number-theory courses worldwide.

Why it matters: The infinitude of primes is the foundation of every modern cryptographic system. RSA encryption works because primes are abundant — large enough primes are always available, and there is no finite list of "the primes worth knowing about." Euclid's proof, written before the wheel was widely deployed for transport, secures every credit-card transaction today.

The story did not stop with Euclid.

Pierre de Fermat (1607–1665, France) introduced the small theorem now named after him, which states that for any prime ppp and any integer aaa not divisible by ppp, ap−1≡1(modp)a^{p-1} \equiv 1 \pmod{p}ap−1≡1(modp).

Bernhard Riemann (1826–1866, Germany) connected the distribution of primes to the zeros of an analytic function — the Riemann Hypothesis, still unsolved, is widely considered the most important open problem in mathematics.

Worked Examples Prime Numbers

Example 1: Test whether 979797 is prime (the wrong path first)

The instinct is to divide 979797 by every integer from 222 to 969696. That works but is wasteful. Stop and use a better rule.

You only need to test divisors up to 97≈9.85\sqrt{97} \approx 9.8597​≈9.85 — so divisors 2,3,5,72, 3, 5, 72,3,5,7 (skipping composites). If none of these divides 979797, the number is prime.

Final answer: 979797 is prime.

Example 2: Prime factorise 848484

84=2×42=2×2×21=2×2×3×784 = 2 \times 42 = 2 \times 2 \times 21 = 2 \times 2 \times 3 \times 784=2×42=2×2×21=2×2×3×7

Or written with exponents: 84=22×3×784 = 2^2 \times 3 \times 784=22×3×7.

Final answer: 84=22×3×784 = 2^2 \times 3 \times 784=22×3×7.

Example 3: Find the next prime after 313131

Test 323232: even, not prime. Test 33=3×1133 = 3 \times 1133=3×11, not prime. Test 343434: even, not prime. Test 35=5×735 = 5 \times 735=5×7, not prime.

Test 363636: even, not prime. Test 373737: not divisible by 2,3,52, 3, 52,3,5, and 37≈6.08\sqrt{37} \approx 6.0837​≈6.08, so we need only check those. 373737 is prime.

Final answer: The next prime after 313131 is 373737.

Common Mistakes of Prime Number

Mistake 1: Treating 111 as a prime number.

Where it slips in: The student reasons that 111's only divisors are 111 and itself, so 111 should be prime.

Don't do this: Include 111 in any prime list.

The correct way: By convention, primes are integers strictly greater than 111. Excluding 111 preserves the unique-factorisation property.

Mistake 2: Treating 222 as composite because it is even.

Where it slips in: The student remembers "even numbers are not prime" and applies the rule to 222.

Don't do this: Skip 222 when listing primes.

The correct way: 222 is the only even prime — its only divisors are 111 and 222. Every other even number has 222 as a third divisor, but 222 itself does not have any "other" even divisor.

Mistake 3: Stopping the divisor test too early.

Where it slips in: Testing divisors 2,3,52, 3, 52,3,5 on, say, 221221221, finding none divide it, and declaring 221221221 prime — but 221=13×17221 = 13 \times 17221=13×17.

Don't do this: Stop at small primes.

The correct way: Test all primes up to N\sqrt{N}N​. For N=221N = 221N=221, 221≈14.87\sqrt{221} \approx 14.87221​≈14.87, so test 2,3,5,7,11,132, 3, 5, 7, 11, 132,3,5,7,11,13. The number 131313 catches the factorisation.

Mistake 4: Confusing prime with odd.

Where it slips in: The student assumes every odd number is prime.

Don't do this: Treat odd as a sufficient condition.

The correct way: Many odd numbers are composite — 9=329 = 3^29=32, 15=3×515 = 3 \times 515=3×5, 21=3×721 = 3 \times 721=3×7. Odd is necessary (after 222) but not sufficient.

The real-world version of Mistake 3 is the RSA challenge problems. RSA Laboratories published large numbers and offered cash for anyone who could factor them. Several were factored by distributed computing networks running for months. The fact that some are still unfactored after decades is exactly why RSA encryption is still in use — the divisor test simply does not finish in any reasonable time at the sizes used in real cryptography.

Try These Next

Now try this: list all primes between 100100100 and 120120120 using the Sieve of Eratosthenes (you only need to test divisors up to 120≈10.95\sqrt{120} \approx 10.95120​≈10.95, so primes 2,3,5,72, 3, 5, 72,3,5,7). If you get stuck, return to the sieve description above.

If your child is comfortable with primes up to 100100100, the natural next step is prime factorisation of larger numbers and modular arithmetic — both lead directly to RSA encryption. At Bhanzu, trainers connect prime numbers to where they actually do useful work, so the topic earns its place in the curriculum.

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Frequently Asked Questions

Is 1 a prime number?

No. By modern convention, primes are integers greater than 111. Excluding 111 preserves the uniqueness of prime factorisation.

How many prime numbers are there between 1 and 100?

Exactly 252525. Listed above.

What is the largest known prime number?

As of 2026, the largest known prime is a Mersenne prime with over 242424 million digits. Mersenne primes have the form 2p−12^p - 12p−1 for some prime ppp, and the search for new ones is ongoing through GIMPS (the Great Internet Mersenne Prime Search).

Why are prime numbers important in cryptography?

Because multiplying two large primes is easy, but factoring the product is computationally hard. RSA encryption builds its security on this asymmetry.

Is there a formula that generates all primes?

No simple formula. Several formulas generate primes for some inputs (Euler's n2+n+41n^2 + n + 41n2+n+41 gives primes for n=0n = 0n=0 to 393939, then fails). The pattern of primes is famously irregular — that irregularity is the subject of the Riemann Hypothesis.

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Bhanzu Team

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