Only numbers that are perfect squares have an odd number of positive factors. All other types of numbers have an even number. However, if negative factors are included, then all numbers have an even number of factors.
Factors occur in pairs because pairs of factors multiplied together produce the factored number. For example, 12 is produced by multiplying 1 and 12, or 2 and 6, or 3 and 4, so 12 has six different factors. With a perfect square, there is always one factor that produces the number when multiplied by itself. For example, 16 is produced by multiplying 1 and 16, or 2 and 8, or 4 and 4, so there are only five different factors for 16.
Learn MoreThe number with the most factors is infinite. The number with the most known factors contains all of the prime numbers, of which Euclid's theorem also suggests there is an infinite number.
Full Answer >The numbers between 30 and 40 whose only factors are one and themselves, also called prime numbers, are 31 and 37. Prime numbers are useful in generating numerical passwords or codes that are difficult to crack.
Full Answer >The following numbers under 100 have the most factors: 60, 72, 84, 90 and 96. They each have a total of 12 factors, including the number one and the number itself.
Full Answer >The factors of the number 23 are 23 and 1. Factors of a number are the numbers that are multiplied together to equal the given number.
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