Math Question?

Question

If you have the prime factorization of a number, how do you predict without multiplying how many factors the number has?

the question doesn't really make much sense to me. but it's exactly what the teacher wrote on the board. it's grade 8 math..

Answer 1

The answer man has explained HOW to produce a prime factorization, but the question your teacher asked is how to predict the amount of factors the number has without multiplying.

In other words, if the prime factorization is 2 x 2 x 3, the amount of factors the number has is 5 - 2, 3, 4, 6 and 12. Of course, 1 is always a factor, so 2 x 2 x 3 has 6 factors in total.

How to predict that? Basically, the prime factors always make up ALL the factors of the specific number. That is to say ALL factors come from prime factors multiplied with other prime factors.

So 2x2x3 has 6 factors. The obvious one is 1.
Step 2 is simple too. Prime factors are also considered factors, so in 2x2x3, you have 2 different factors - 2 and 3.
Step 3, you multiply two of the numbers to create new factors. Two more factors arise - 2x2 and 2x3 (4 and 6).
Step 4, you multiply three of the numbers to create new factors. The final factor arises - 2x2x3, which is 12.
Step 5, you multiply four of the numbers etc etc. (if applicable)
Step 6, you multiply five of the numbers etc etc. (if applicable)
It goes on and on!

Therefore there are 6 factors all derived from the prime factorization of 2x2x3.

Now you can estimate anything. For example, if i gave u the prime factorization 5x3x2, you can follow my steps.
Step One = 1 new factor
Step Two = 3 new factors
Step Three = 2 new factors
Step Four = 1 new factor.

Now you know how to predict how many factors the number has without multiplying.

Answer 2

I don't think you can predict the number of prime numbers. I think the easiest way to do this is to write the number, pick a low prime number and write it to the left, draw an upside down division, write the result of dividing by that prime number, and continuing the process until all factors are known. Here is a little piece on factoring.

Factoring Numbers

"Factors" are the numbers you multiply to get another number. For instance, the factors of 15 are 3 and 5, because 3×5 = 15. Some numbers have more than one factorization (way of being factored). For instance, 12 can be factored as 1×12, 2×6, or 3×4. A number that can only be factored as 1 times itself is called "prime". The first few primes are 2, 3, 5, 7, 11, and 13. The number 1 is not regarded as a prime, and is usually not included in factorizations, because 1 goes in to everything. (The number 1 is a bit boring in this context, so it gets ignored.)
You most often want to find the "prime factorization" of a number. That is, you usually want to find the list of all the prime-number factors of a given number. The prime factorization does not include 1, but does include every copy of every prime factor. For instance, the prime factorization of 8 is 2×2×2, not just "2". Yes, 2 is the only factor, but you need three copies of it to multiply back to 8, so the prime factorization includes all three copies.

On the other hand, the prime factorization includes ONLY the prime factors, not any products of those factors. For instance, even though 2×2 = 4, and even though 4 is a divisor of 8, 4 is NOT in the PRIME factorization of 8. That is because 8 does NOT equal 2×2×2×4! This accidental over-duplication of factors is another reason why the prime factorization is often best: it avoids counting any factor too many times. Suppose that you need to find the prime factorization of 24. Sometimes a student will just list all the divisors of 24: 1, 2, 3, 4, 6, 8, 12, and 24. Then the student will make the product of all these divisors: 1×2×3×4×6×8×12×24. But this equals 331776, not 24! So it's best to stick to the prime factorization, even if the problem doesn't require it, in order to avoid either omitting a factor or else over-duplicating one.

In the case of 24, you can find the prime factorization by taking 24 and dividing it by the smallest prime number that goes into 24: 24 ÷ 2 = 12. (Actually, the "smallest" part is not as important as the "prime" part; the "smallest" part is mostly to make your work easier, because dividing by smaller numbers is simpler.) Now divide out the smallest number that goes into 12: 12 ÷ 2 = 6. Now divide out the smallest number that goes into 6: 6 ÷ 2 = 3. Then the prime factorization is 2×2×2×3.