## Sunday, April 7, 2013

### Is 247 prime?

Prime and composite numbers

Questions:  Today’s post is about prime and composite numbers.   Is 247 a prime or composite number?  How can we efficiently determine whether a number is prime or composite?

Answers:  247 is a relatively large numbers but it is not difficult to quickly determine whether numbers of this size have any other factors than one and itself.

247 is not divisible by 2 or any even number.

Any factor of 247 must be less than half of 247 and greater than 2.

247 is not divisible by 3.  (the sum of the digits in not divisible by 3 so the number itself is not divisible by 3.)   See the link below on the divisibility rules.

We now know that any factor of 247 must be between 3 and one-third of 247.

247 is not divisible by 5 because the ones digit is not 0 or 5.  Again see the divisibility rule link.

247 is not divisible by 7.  I determined this by dividing.  (Note there is a divisibility rule for 7.  Seems interesting and may be the subject of another post.)

Any possible factor of 247 must be between 7 and one seventh of 247 (32.)   We are making progress!!!

9 is not divisible because 3 is not divisible.

11 is not divisible. 11 goes into 247 21 times with a remainder of 6.

13 is divisible.  (13 x 19  = 247).

Even though 247 is a fairly large number the process of finding factors by process of elimination (or the process of finding there are no factors) goes really quickly.

One more point:  Note that 247 is the product of two primes.  Intuitively, when a number is a product of two primes the two primes are the only factors of the number excepting of course 1 and the number.

The factors of a number can be found by rearranging the prime factors.  For example, 12 can be written as 2x2x3.  This means the factors of 12 are 2 and 2x3 and 2x2 and 3.

Since 247 is equal to the product of two primes further regrouping is impossible.

There are a lot of resources on prime numbers on the web.  Some of best explanations come from Khan academy.  See the link below:

I hope my post is of some use.