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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

Q: How do you use prime factorization to find out how many factors it has?

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As a product of its prime factors: 3*7*7 = 147

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