why is finding the prime factorization of a number useful?

Answer 1

Once you have the prime factorization of a number, it's easy to find the Greatest Common Factor (GCF) and the Least Common Multiple (LCM), both of which are necessary for doing fractions.

There's an even more important reason that most people don't know about, though.

On the internet, when you buy something using a credit card, it would be REALLY easy for someone to get your credit card and personal information unless it is "encrypted". Encryption is the process of encoding a message so that it's very hard for someone else to read it.

It turns out that the most powerful form of encryption available to us requires the use of very very large prime numbers.

Answer 2

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.

Answer 3

The prime factorization of a number is useful for finding the lowest common multiple. Lowest common multiple is important when adding fractions.

For instance, suppose we wanted to add 4/9 + 5/24. We would require the lowest common multiple of 9 and 24, or lowest common denominator.

To calculate the lowest common multiple, we need the prime factorization of 9 and 24.

9 = 3 x 3
24 = 3 x 8 = 3 x 2 x 4 = 3 x 2 x 2 x 2 = 2 x 2 x 2 x 3

Now, we place these prime factors next to each other, top to bottom, keeping in mind common factors

9 = ||||||||||||||||||||||||||3 x 3
24 = 2 x 2 x 2 x 3

After they're lined up, we form a new number based on them lining up with their common factors

2 x 2 x 2 x 3 x 3 = 72

So our common factor is 72

4/9 = ?/72
5/24 = ?/72

since 9 x 8 = 72, we multiply the 4 by 8 to get 32. So
4/9 = 32/72

Since 24 x 3 = 72, we multiple the 5 by 3 to get 15. So
5/24 = 15/72

Now, we can add the numbers up normally:

32/72 + 15/72 = 47/72

As you can see, prime factorization plays a HUGE role when adding and subtracting fractions, because we needed to get the lowest common denominator.

Answer 4

its not

Answer 5

its not