Q&A for How to Find How Many Factors Are in a Number

There are several methods for factoring a number. One approach is to identify its factor pairs, such as with the number 20, where you find pairs like 1 times 20, 2 times 10, and 4 times 5. It's important to check each prime number initially and stop when you encounter repeated factors. Another method involves prime factorization, where you break the number into its prime factors. Taking 20 as an example again, you can break it down into 4 times 5. Since 5 is prime, it is circled and left untouched, while 4 is further broken down into the prime factors 2 and 2. Once all numbers are expressed as their prime factors, the factoring process is complete.

Divide the number by the smallest prime number above 1 that will fit into it and give you a whole number result (with no remainder). Do the same with the resulting number, and keep doing it until you reach the answer “1.”

This is a difficult question to answer, since there is an infinite number of numbers. Every number has at least 2 factors (1, and the number itself). Some numbers, known as “highly composite numbers,” can have very large numbers of factors. For example, 840 has 32 factors. For more information, check out the work of Achim Flammenkamp, who has created an algorithm for finding highly composite numbers.

If you have a graphing calculator, you can do this by entering the equation y = [your number]/x and then finding all the whole number pairs of x’s and y’s in the resulting table or graph. You can also do a search for “factor calculator” online if you don’t have access to a graphing calculator.

Any whole number (greater than 1) which is exactly divisible only by itself and the number 1.

Each of those numbers has two factors: the number itself and 1. That makes each a prime number.

3 x 97² is the prime factorization. The prime factors are 3 and 97.

A prime number has only two factors -- 1 and itself. This is true of all prime numbers.

Just find all of the square numbers. Since they are square, that means that they have 2 of same number, otherwise it would just have pairs of numbers.

1, 3, 5, 7, 9, 11, 13, 15, 17, etc.

4 has three factors: 1, 2, and 4.

Suppose you were looking for a factor of 120, which factors as 2^3 × 3^1 × 5^1. Unique factorization means that any divisor of 120 can't have any prime factor other than those, nor use a higher power of those primes than 120 has. So that divisor must itself factor as 2^a × 3^b × 5^c where a = 0,1,2, or 3 b =0 or 1, c= 0 or 1.

Technically the number 1 has two factors, 1 and 1 ("1 and itself"). In practical terms, however, you could say that 1 has just one factor, that being 1.

Test each prime number smaller than the number being factored to see if it divides evenly into the number. If so, it's a factor.

60 = 2² x 3 x 5.

Yes. 1, 2, 4, and 8. (We're not referring to prime factors in this case, just factors.)

Follow the steps in Do a Factor Tree .

Two is the first positive prime number. One is not prime, and two is only divisible by itself and one.

Convention is to consider only the positive factors. If you did want to count negative factors as well, just double it. But the reason for the convention is that this function has nicer properties if you don't. For example, if a and b have no common factors, then d(ab) = d(a)*d(b).

Odd numbers cannot be divisible by 2, but they may be divisible by other numbers. For example, 9 is divisible by 3. If it is not divisible by any number besides 1 and itself, the number is a prime number.

Do each number individually. Follow the steps above and create a factor list for each number.

The best way to find the prime factors of any number is to divide the number repeatedly by the smallest prime numbers (other than 1). Start with 2. Divide repeatedly by 2 until you can't do it anymore. Then divide by 3 until you can't divide anymore. Then divide by 5, if possible, and by 7, if possible. At this point you will probably have found all the prime factors. (If not, try dividing by 11 or 13. At that point--or before--you should have all the prime factors.) In the case of 30, dividing repeatedly by 2, 3 and 5 will find all the prime factors.

Express your number as powers of its prime factors. E.g. 40 = 5^1 x 2^3 Add 1 to each power and mulpily 1+1 = 2, 3+1 = 4 Number of factors is 2 x 4 = 8. Proof - look at the combinations of each factor's power 5^0 = 1 and 5^1 = 5, 2^0 = 1, 2^1 = 2, 2^2 = 4 and 2^3 = 8 i.e (1 + 5 ) x ( 1+ 2+ 4+ 8) each of the 8 terms in the expansion is a factor.

There are essentially 3 methods. 1.) There are tables of factors (also called factorization tables) that list factors of a number. So you look up 899 in one of these tables and find that it's 29*31, both of which are prime. 2.) You could generate the data for these tables yourself. Look up "Sieve of Eratosthenes". This works for "small" numbers but not for large ones. (Note that RSA public key cryptography relies on the fact that factoring large numbers is hard. Currently "large" is 2^1048 [approximately 1 followed by 300 0s] to 2^2096.) 3.) Sometimes we can see a pattern. 899 = 900 - 1 = 3^2 - 1 = (30 - 1)(30 + 1). So 4.

It depends on what you want to do. If you are doing school homework, then the teacher probably wants you to do trial division. So try dividing small primes into the number (let's do 81675 for an example). We don't have to try 2 because it's not even. We can try 5. (We know that will work because it ends in 5.) So 81675 = 5^2 * 3267. Then 3 divides 3267 (the sum of the digits is divisible by 3), and we find 3267 = 3^3 * 121 = 3^3 * 11^2. Thus 81675 = 5^2 * 3^3 * 11^2. For large numbers, factorization is considered a hard problem. Google "RSA public key cryptography". This is what (some) mathematicians work on today.

The factors of 66 are 2, 3 and 11. All three of those numbers are prime.

Only prime numbers are cited as factors. The only even prime number is 2. All other factors will be odd.

If you're asking about the greatest factor common to all four of those numbers, it's 3.

Any number divisible by 10 is also divisible by both 2 and 5. So just find the numbers divisible by 10. That means the lowest number you're looking for is 10, and all the other numbers are multiples of 10. There are 15 multiples of 10 from 10 up to 150.

Many numbers have factors without exponents. That's not unusual. Just list them individually with all the other factors. For example, the factors of 63 are 3² and 7.