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Prime Numbers and Internet Security, Explained Simply

Published October 3, 2026Written and reviewed by Mateuss M.

Prime Numbers and Internet Security, Explained Simply — illustration
In short: Multiplying two primes is easy; recovering them from the product is hard. Take 61 and 53: their product is 3233, and factoring 3233 gives 3233 = 53 × 61.

Easy one way, hard the other way

Multiplying 61 × 53 takes seconds by hand. Going backwards — given only 3,233, find its prime factors — means testing divisors. For small numbers that is quick, and a computer does it instantly.

Factoring 3,233
Answer3233 = 53 × 61
  • Divisors: 4
  • Sum of divisors: 3348
Step-by-step working 3 steps
  1. 1
    Divide by the smallest prime that fits, again and again
    3233 ÷ 53 = 61
61 ÷ 61 = 1

    Stop when the quotient reaches 1.

  2. 2
    Collect the primes
    3233 = 53 × 61
  3. 3
    Count the divisors
    (2 × 2) = 4 divisors

    Add 1 to each exponent and multiply.

Scaling it up

Real systems use primes hundreds of digits long. Multiplying them is still instant, but no known method factors such a product quickly on today’s classical computers. That gap is the foundation of RSA, one of the first public-key cryptosystems.

This is an introduction, not a security guide. Real cryptography adds many more safeguards and should never be implemented by hand.

Try it yourself

Use the prime factorization calculator on numbers of increasing size and notice how the trial divisions grow. Doubling the number of digits does far more than double the work.

Frequently asked questions

Are all numbers a product of primes?

Yes. Every whole number above 1 has exactly one prime factorization, apart from the order of the factors.

Can quantum computers break this?

A large enough quantum computer running Shor’s algorithm could factor such numbers efficiently, which is why new post-quantum schemes are being standardised.

Written and reviewed by Mateuss M.

Mateuss M. writes and reviews mathematical content for CalcSolver, focusing on online calculators, formulas, equations, and practical math tools. He reviews calculator functionality, calculation methods, formulas, examples, and explanations to help ensure that each tool is clear, useful, and easy to understand.

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