Prime Factorization (with examples)

It's Time for Math!
It's Time for Math! | Source

Mathematics is a universal language of science. It is used in completely different fields such as natural science, engineering, social sciences, medicine and many others. New branches of mathematics appear over the time. Some of these new disciplines are statistics or game theory. Number theory (or simply arithmetic) is quite old branch of pure mathematics. Number theory studies integers, such as prime numbers. Also it analyses objects that are made out of integers such as rational numbers. Prime factorization is a branch of number theory. It analyses decompositions of a composite numbers into smaller divisors. These divisors, when multiplied together gives result that is equal to the original number.

Prime numbers

Prime number (or simply prime) is a natural number (positive integer) that has no positive divisors other than 1 and itself. Prime number must be greater than 1. The prime numbers up to 1000 are shown below.

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997

Examples:

Question 1: Determine if number 1000 is a prime number or not?

Answer. Number 1000 is not a prime number because it can be divided by 1, 2, 4, 8, ..., 100, 200 and other numbers.

Question 2: Determine if number 1009 is a prime number or not?

Answer. Number 1009 is a prime number because it can be divided only by 1 an by 1009.

Factorization

Factorization is decomposition of a number, polynomial or other object into a product of other objects. These objects are factors. The result of multiplication of factors will be an number, polynomial, matrix or other object.Prime factors are factors that are prime numbers.

Example:

15 = 3 * 5. It can be said that 15 factors into primes as 3 * 5.

Prime Factorization

Prime factorization is a process of finding which prime numbers can be multiplied together to get the original number. It is obvious, that all integers can be factorized into the prime numbers, because prime numbers can be called a basic building block of natural numbers.

Prime factorization algorithm:

1. Find the smallest prime number that divides original number without a remainder.

2. Take the result and find smallest prime number that divides the result without a remainder.

3. Repeat until you will get the result that is prime number.


Example:

Let's find prime factors of number 80. The smallest prime number that divides 80 without a remainder is 2. The result of division is 40. The smallest prime number that divides 40 without a remainder is also 2. The result of division is 20. The smallest prime number that divides 20 without a remainder is 2. The result of division is 10. The smallest prime number that divides 10 without a remainder is 2. The result of division is 5. 5 is a prime number it self. So the final result of prime factorization of 80 would be: 80 = 2 * 2 * 2 * 2 * 5.


Any natural number has only one unique set of prime factors

Factoring the Time
Factoring the Time | Source

Why prime factorization is so important?

As it was mentioned, prime numbers are building blocks of all natural numbers. This property is very useful when working with big numbers. You may ask how? For example, let's talk a little bit about cryptography. Cryptography is very important field in our modern society. It can be said, that who controls the information, controls the world. That is why security of our data is so critical. Cryptography is used everywhere: in computing, internet business, internet banking and so on. Cryptography basically speaking is a field that studies secret codes. People, who create different codes and systems based on numbers works with prime numbers and factorization. Prime factorization may seem quite easy operations, but assume that you have a very large number. To find prime factors for such a number will take a very long time even using a computer. It means, that such a code cannot be broken.

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