If you know the prime factorization of a number, you can determine the number of factors it has by looking at the possible combinations of the factors.
If the prime factorization of a number is p1a * p2b *p3c , then its factors (including one and the number itself) will p1i * p2j *p3k , with i a whole number between 0 and a, j a whole number between 0 and b, and k a whole number between 0 and c.There are (a+1) numbers that i can be, (b+1) numbers that j can be, and (c+1) numbers that k can be, so there are (a+1)*(b+1)*(c+1) factors (including one and the number itself).
It makes more sense using a concrete example:
Consider the number 360. We want to figure out how many factors it has.
It's prime factorization is 23*32*5. For a number to be a factor of 360, it will have to have only these prime factors, and the largest exponent the exponent can be for each of the factors is that factors exponent in the prime factorization of 360. So a factor of 360 will be a product of 2 (to the 0,1,2, or 3) power, 3 (to the 0,1, or 2), and 5 (to the 0 or 1 power). There are thus 4*3*2 = 24 factors of 360 (including 1 and 360). We can check this by systematically listing out the factors of 360:
20*30*50=1
20*30*51=5
20*31*50=3
20*31*51=15
20*32*50=9
20*32*51=45
21*30*50=2
21*30*51=10
21*31*50=6
21*31*51=30
21*32*50=18
21*32*51=90
22*30*50=4
22*30*51=20
22*31*50=12
22*31*51=60
22*32*50=36
22*32*51=180
23*30*50=8
23*30*51=40
23*31*50=24
23*31*51=120
23*32*50=72
23*32*51=360
Two distinct prime factors, six total prime factors.
As a product of its prime factors: 3*7*7 = 147
You can find prime factorization worksheets at math world dot com. They have many to choose from and download and print for your convenience. Great site to use.
Many people find factor trees to be the easiest.
Every prime number has exactly 2 factors, 1 and the number itself.
Two distinct prime factors, six total prime factors.
Divide it by each prime number which does not have remainder.
There are 2 prime factors of 48. The prime factorization of 48 is: 24 x 3 = 48
To find a prime factorization, divide a composite number and its factors by prime numbers until all the factors are prime. Many people find factor trees helpful in visualizing this process. Example: 210 210 Divide by two. 105,2 Divide by three. 35,3,2 Divide by five. 7,5,3,2 Stop. All the factors are prime. 2 x 3 x 5 x 7 is the prime factorization of 210.
As a product of its prime factors: 3*7*7 = 147
You can find out the number of factors of any number from its prime factorization by adding one to each exponent and multiplying. The prime factorization of 156 is 2^2 x 3^1 x 13^1 3 x 2 x 2 = 12 156 has 12 factors.
The prime factorization of 420 is: 2 × 2 × 3 × 5 × 7
Zero has infinitely many factors, so a prime factorization would be unwieldy and ultimately meaningless. The prime factorization of 48 is 2 x 2 x 2 x 2 x 3
For composite numbers, only one string of factors is the longest; the prime factorization.
You can find prime factorization worksheets at math world dot com. They have many to choose from and download and print for your convenience. Great site to use.
Many people find factor trees to be the easiest.
3 and 7 are the two different prime factors of 63. Prime factorization of 63 is 3 x 3 x 7.