Notes on Factors, Prime Numbers, and Prime Factorization
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:
• All numbers are divisible by 1
• Any even number is divisible by 2. A number is divisible by 2 if it ends with a 0, 2, 4, 6, or 8.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 4 if the last two digits (tens and ones) are divisible by 4.
• A number is divisible by 5 if it ends in a 5 or 0.
• A number is divisible by 6 if it is also divisible by 2 and 3 (see these tests above).
• To test a number for divisibility by 7
o Take the last digit in a number.
o Double and subtract the last digit in your number from the rest of the digits.
o Repeat the process for larger numbers.
• A number is divisible by 8 if the last 3 digits are divisible by 8.
• A number is divisible by 9 if the sum of its digits are divisible by 9
• A number is divisible by 10 if it ends in a zero.
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.
