The factors of 65 are one, five, 13 and 65. These are the only natural numbers that have 65 as a product when multiplied by one or more other numbers.
Except for the number one, any given natural number will have itself and one as factors, so one and 65 are the easiest factors to identify. The remaining factors are found by attempting to divide 65 by prime numbers, from least to greatest, until halfway to 65.
A prime number is a number whose only factors are one and itself, so the smallest prime number is two. Since 65 is an odd number, it cannot be divided by two, which eliminates two as a factor and any multiple of two.
The next prime number is three. However, a number is not divisible by three if the sum of its digits cannot be divided by three without a remainder. In that 6 + 5 = 11, and 11 cannot be divided by three evenly, 65 cannot be divided by three either or any multiple of three.
Five is the next prime number and any number that ends with five or zero can be divided by five. Five must therefore be a factor of 65. Its corresponding factor is 13.
The only prime numbers that remain between 13 and halfway to 65 are 17, 19, 23, 29 and 31. However, none of these primes divides into 65 without a remainder, so it is safe to conclude that the list of 65’s factors is complete.