How To Determine the Number of Divisors of an Integer
A factor is a number that divides into a larger number.It is easy to determine how many divisors a small number has by listing out all the different ways you can get to that number.Finding the number of divisors is more difficult when working with larger numbers.You can use a simple formula to reach your answer once you have factored the integer into prime factors.
Step 1: Write the number at the top of the page.
You have to leave enough room for a factor tree to be below it.You can factor a number using other methods.For more instructions, read Factor a Number.If you want to know how many divisors the number 24 has, you can write it at the top of the page.
Step 2: You don't have to include 1 if you find two numbers to get the number.
There are two factors of the number.Write the two factors below a split branch from the original number.Draw a split branch from 24 and write the numbers 12 and 2 below it.
Step 3: Prime factors should be looked for.
A prime factor is a number that is evenly divided by itself.7 is a prime number because the only numbers that evenly divide into 7 are 1 and 7.Keep track of any prime factors by circleing them.If 2 is a prime number, you would circle it on your factor tree.
Step 4: Factor in non-prime numbers.
When all of your factors are prime, keep drawing branches down from the non-prime factors.To keep track of the prime numbers, circle them.12 can be factored into 6 and 2 styles.You would circle it since it is a prime number.6displaystyle 6 can be factored into 3 and 2 displaystyles.You would circle them since they are prime numbers.
Step 5: Each prime factor has an exponential expression.
Look for multiples of the prime factors in your factor tree.The number of times a factor appears is the same as the percentage of the factor in the expression.The exponential expression is 23displaystyle 23, because the prime factor 2 appears three times in your factor tree.The prime factor is displayed 1 time in your factor tree.
Step 6: The prime factorization of the number requires an equation to be written.
The product of the exponential expressions is equal to the original number you are working with.24=23 times 31 is a displaystyle.
Step 7: Determine the number of divisors, or factors, in a number with an equation.
D(n) is equal to the number of divisors in the equation.You could have less than three or more.The formula states to work with any number of exponents.
Step 8: The formula can be plugged in with the value of each exponent.
The prime factors should not be used for the exponents.You would use the exponents 3displaystyle3 and 1 displaystyle1 into the equation.The equation will look like this.
Step 9: The values should be added in parentheses.
Adding 1 to each exponent is all you have to do.For example, d(24) is a displaystyle.
Step 10: Put the values in parentheses.
The number of factors will be equal to the product's number.The number of factors in the number 24 is 8.