Expert: Josh - 8/22/2006

Question
im filling a study guide in and i forgot my book at school. i dont even get what to do or how to do the math quetion.HELP THANKS

write 1389 as the product of its prime numbers.  

Answer
Hi Madison,

The question relates to a result which states that any integer (whole number) can be factorized as a product of prime numbers.

To actually obtain the prime factorization, we first write down a list of the prime numbers. We denote the set of prime numbers by P={2,3,5,7,11,13,17,19,23 etc.}

Recall that a prime number, by definition, cannot be divided by any other integer, except the number itself and one. The integer "1" is NOT considered a prime number.

Given an integer N, we obtain the prime factorization by reducing this number. First, we try dividing N by the smallest prime in the list P, which is 2 here. If N does not divide nicely into 2, we move on and try the next larger prime number in the list. Does 3 divide 1389, well 1389=3*463. There is no remainder. So, 3 divides 1389.

We write 1389 = 3 * 463.

Next, we let N=463 and continue to find the remaining prime factors in a similar fashion.

Again, we start with the smallest prime in the list P. viz., p=2. Does 2 divide 463? No, so try the next larger prime in the list. Does 3 divide 463? Well, 463=3*154+1. The remainder (1) is non-zero. So, we have to try the next larger prime number, 5. Does 5 divide 463. We should remember that only numbers ending with "5" or "0" in the last digit can be divided by 5. So, again, 5 fails to divide 463. We keep going, so the question is when do we stop. How can we tell if 463 is not a prime number itself.

It turns out that 463 is itself a prime number. We know this only when we try all the prime numbers (p) in the list P, in increasing order, until p is larger than the integer rounded up from the square root of N.

e.g., Here, the number under consideration is N=463. Its square root, sqrt(N)=21.5174 approximately. So, we should try dividing N by p=2, p=3,....p=23 and stop if all fails. Note that p=23 is the next prime number just above 21.5174. When all prime numbers from p=2 to p=23 fail to divide 463 -- this is the case here -- we declare 463 a prime number.

So, answer to your question (the prime factorization) of 1389 is 3*463.

Exercise: Try finding the prime factoriztion for 168 using this method. You should be able to show that 168=2*2*2*3*7

Hope this helps:)