Prime Factors

Though multiplication and its inverse, division, are performed easily with all whole numbers, the principle of equivalency leaves an easier solution to working with many of them.  The practice of "finding the factors" need not end with big numbers. In taking on large numbers, it is perfectly alright to break them up into the numbers from which they came by way of multiplication.

Even Numbers.

One half of all natural numbers are "even," this is to say they can be divided by the number 2.  Two is the first "prime" factor.  Almost everyone remembers the cheer: "Two, four, six, eight, who do we appreciate?"  Well, those are the first four "even" numbers, which hypothetically go on forever.  Along with the the beginning whole number "0," these provide the clue that a number has at least three factors: 1, 2 and the number in question.

Whether you multiply an even or an odd number by an even number, the answer will be even.

For example:  1234 is an even number, and therefor is not a prime number.  Its factors are 1, 2 and 617.  Is 617 a prime number?  Well, it isn't even.  Two is the only "even" prime number, so let us move on to other "prime suspects"

Odd Numbers

The other half of all natural numbers are odd.  This does not mean they are prime, but it makes task of finding big prime numbers a little easier.  I am not one to pursue such a task.  Suffice it to say that an odd number needs to be approached with care.  It can have hidden factors just waiting to be discovered.

Taking the factor of 1234 -- 617 -- the first thing is to find that number's square root.  This is best done with a calculator, but I recognize this as close to 25x25, that is 625.  This sets the limit.  A prime factor would have to be under 25, but not by much.  23x23 has a product 529.

Dropping back to the basics, then, we start with 3.  By theorem, the sum of the digits of any number must be divisible by 3 if the number is divisible by 3. 6+1+7 = 14.  14 is not divisible by 3.

The next prime number is 5. "Counting by fives" is easy, and it reveals to numbers, one even and one odd.  Every even number ending in 0 is divisible by 2 and 5.  This is two for one!  So, 617 is not divisible by 5 either.

The last odd number under 10 is the number 7.  There is no easy way to tell if a number is divisible by 7.  In the case of 617, we see a seven, but the first two numbers return a remainder of 5, leaving 57 (not divisible by seven.  Knowing the multiples of 7 up to at least 9x7 is advisable.

Note that multiplying an odd number by an odd number will get an odd number:

1x1 = 1 5x5 = 25 9x9 = 81
3x3 = 9 5x7 = 35
3x5 =15 5x9 = 45
3x7 =21 7x7 = 49
3x9 =27 7x9 = 63

With the factors 11, 13, 17, 19, and 23, only 11x17 even remotely comes close. However though 11x17 ends in 7, it is far from 617. What about 11x27?  That gets closer, but is far short as well (270 + 27 = 297).

So, 617 is indeed prime.

The most important products to know are those of the prime numbers 2, 3, 5 and 7.  Note, standing by itself is the number 49!  The "new" answer to the universal question!

2|  4

3|  6  9

5| 10 15 25

7| 14 21 35 49

      2  3  5  7 

Finding prime factors:

Starting with 2, what are the prime factors of 7,984,356 (a totally random seven digit number!)

Immediately 2 "works."  Trying 4, we get 1,996,089. Not even, so on to 3.  These digits are 1+9+9+6+0+8+9.  Added this gives us 42; reducing further to 6

So, with factors 2x2x3, we can divide by 12 to get 666,563  Is ths as far as we can go?  Not divisible by 5, so we try 7, 11, 13 and 17.  Seventeen works, yielding 39,139.

This leaves factors of 1, 2, 3, 4, 6, 12, 17, and 39,139.

Using a handy calculator, I know that the square of that large number is just under 198.  197 is a prime number having as its square 38,809. 197 times 199, the next prime number, equals 39,203. This means there are no more prime factors of our chosen number.

The prime factors of 1,996,089 are 2,3,17 and 39,139.


So, what do I know?

Odd x odd = Odd number
Odd x even = Even number
Odd + even = Odd number
Odd + odd = Even number

A prime number is a natural number that has exactly two natural divisors: 1 and itself.

2, 3, 5 and 7 are prime numbers under 10.

Even number 2 is the powerhouse of the primes, affecting ALL even numbers.

Odd number 3 can be seen to be a factor if the sum of the digits add up to a number divisible by 3.

Odd number 5 shouts out from the end of one of its products.  If the even number 0 is there, the 2 and 5 are instantly known.

Seven times seven is forty-nine (7x7=49). Every digit between 1 and 9 shows up as the final digit of multiples of 7.  So, don't look for an easy out here.

All prime numbers larger than 2 are odd numbers. About one in four numbers is prime (in the first 200 natural numbers, at least).

The factors of any number start with one and end with the square root of that number.