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Just knowing the divisibility rules for the first four prime factors (2, 3, 5 and 7) will help find the prime factorizations of a large percentage of the numbers you will encounter. At the very least, dividing your original number by those factors should cut it down to a manageable size. The first thing you do when starting a prime factorization is notice whether the number is even. If it is, you can take out two as a factor. If not, you can skip over it. The same with 3 and 5. If you know they are not factors just by looking at the number, it saves a lot of trial and error.

Q: How can you use divisibility rules to help you find the prime factorization of numbers?

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The divisibility rules for a prime number is if it is ONLY divisible by 1, and itself.

You can test successive prime numbers to see if your number is divisible by them, but knowing the divisibility rules will help you eliminate some steps, depending on what your number is. If your number is odd, you don't have to test for 2. If the sum of your number's digits do not total a multiple of 3, you don't have to test for 3. If your number doesn't end in a 5 or 0, you don't have to test for 5. Just by looking at your number, you can include or eliminate the three most common primes if you know the rules of divisibility.

they have to be composite

350 is a composite number. it can be divisible by 10, 35, 5, 2 etc. You will find it easier to tell between prime and composite if you know the divisibility rules. one simple one is that if a number ends with zero it is divisible by 10...there are a lot of other divisibility rules.

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The divisibility rules will show that 53 is not divisible by anything other than 1 and itself. Since it is already prime, it doesn't have a factorization.

The divisibility rules for a prime number is if it is ONLY divisible by 1, and itself.

With the common divisibility rules, you can quickly see that it is divisible by 5, and by 9 (3 x 3). If you divide 225 by each of these numbers, you should be able to get the remaining factors quickly, as well.

3 x 3 x 31 = 279

they can help you by finding the two factors of the number given

By the rules of divisibility, you know that 515 is divisible by 5. 515 divided by 5 is 103. Since both 5 and 103 are prime and can't be divided further, stop there. The prime factorization of 515 is 5 x 103.

The answer will depend on the divisibility rules list.

For any practical purpose, it is easier to simply divide, instead of looking for fancy divisibility rules. However, you can apply the divisibility rules for 3 and for 7. This works because (a) their product is 21, and (b) these numbers are relatively prime.

You can test successive prime numbers to see if your number is divisible by them, but knowing the divisibility rules will help you eliminate some steps, depending on what your number is. If your number is odd, you don't have to test for 2. If the sum of your number's digits do not total a multiple of 3, you don't have to test for 3. If your number doesn't end in a 5 or 0, you don't have to test for 5. Just by looking at your number, you can include or eliminate the three most common primes if you know the rules of divisibility.

they have to be composite

yes I believe 728393 is prime because you have to do all the divisibility rules out!

Suppose you were trying to find the prime factorization of 123. You know that half of the divisors will be less than the square root. Since the square root is between 11 and 12, you only need to test 2, 3, 5, 7 and 11 as prime factors. If you know the rules of divisibility, you already know that 2 and 5 aren't factors and 3 is. It saves time.