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An in-depth exploration of prime numbers and prime factorization. Prime numbers are numbers with exactly two factors: 1 and itself. The smallest 168 prime numbers and explains how to test the primality of a number using divisibility rules. Additionally, the document explains how to find the prime factorization of a number through factorization and factor trees.
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Factors are the numbers that multiply together to get another number.
A Product is the number produced by multiplying two factors.
All numbers have 1 and itself as factors.
A number whose only factors are 1 and itself is a prime number. Prime numbers have exactly two factors.
The smallest 168 prime numbers (all the prime numbers under 1000) 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, 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
There are infinitely many prime numbers. Another way of saying this is that the sequence of prime numbers never ends.
The number 1 is not considered a prime. The definition of a prime number is one that has exactly TWO factors: itself and 1. So the number 1, having only ONE factor, itself, does not meet the definition.
A number with three or more factors is a composite number.
Divisibility rules can be used to factor a number and to test the primality of a number. Some divisibility rules:
A prime factorization is a factor string expressing a number as the product of only prime factors. Every number has exactly one prime factorization. This prime factorization can be written using exponents if any of its prime factors appear more than once in the string.
To factor a number
Example:
36 {1, 36}
36 {1, 2, 18, 36}
36 {1, 2, 3, 12, 18, 36}
36 {1, 2, 3, 4, 9, 12, 18, 36}
36 {1, 2, 3, 4, 6, 9, 12, 18, 36}
In the example above, 5 is not a factor of 36 so it is not in the factor list. 6 is in the middle so stop there, because 7 is not a factor of 36, nor is 8. The next number, 9, is already in the list, and every number greater than 9 has been included or eliminated when the lower factors were added.
To find the prime factorization of a number with a factor tree
Example:
84 84 is the number to be prime factored
2 42 2 is the smallest prime factor of 84 along with 42
2 21 2 is the smallest prime factor of 42 along with 21
3 7 3 is the smallest prime factor of 21 along with 7
7 is prime so stop. The prime factorization of 84 is:
2 x 2 x 3 x 7
To write the prime factorization using exponents, combine like prime factors: 2 2 x 3 x 7
Another way to get the factors of a number is to list its factor pairs using a systematic approach:
1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6
So the factors of 36 are:
1, 2, 3, 4, 6, 9, 12, 18, 36