Prime Factors: Part 4

Prime Factors:

The Royal Sevens

It is said by those who study such things that the number seven is the number of completion.  This is, of course, based on the creation of the world being finished in six days and God resting on the seventh day.  I accept this, for it was codified along with the fourth commandment on tablets of stone.

It is for this reason that I have decided to call the related primes ending in the number 7 the "Royal Family" of primes.  Seven is a combination of three and the first square number, that is, the number 4.

However, the important feature is the relationship with the number 5.  It just so happens that 7 is a "mirror image" of 3. Where 3 is two less that 5, 7 is two more.  This translates into a chart very much like that of the threes.

However, in this case, subtraction back to the 5 will get the answer we need. I will start with 17 and backtrack back to 7.

What relationship does 1 have to 7 to get a something divisible by 17.  It so happens that 5 times 7 is 35 and 2 times 17 is 34.  That looks good, since they are just 1 apart.

The bigger number is the 35, so the operation will have to be subtraction.  It will look odd, but the multiples in our present format will all be negative integers.  But, then I know that -1 is just as good a factor as +1.

Here is the chart (with seven as "07" inserted)

07 => 0-(7*2) = 0-14  =  -14
17 => 1-(7*5) = 1-35  =  -34
27 => 2-(7*8) = 2-56  =  -54    
37 => 3-(7*11)= 3-77  =  -74
47 => 4-(7*14)= 4-98  =  -94
57 => 5-(7*17)= 5-119 = -114
67 => 6-(7*20)= 6-140 = -134
77 => 7-(7*23)= 7-161 = -154
87 => 8-(7*26)= 8-186 = -174
97 => 9-(7*29)= 9-203 = -194

Just like with the threes in the mirror, the multipliers increase by 3.  In this case, they start with 2 instead of 1, so the "mirror" is a bit warped.  The columns are not as neat as the numbers on the left don't match those on the right.  We actually have to do some math to see what is happening here.

When you put a candidate for division in the left column the format makes more sense.  However, with the multipliers getting into the twenties, we need to step back and concentrate on getting our math right.

Let's pick 37 as our prime.  I will use a very simple multiple of 37 to illustrate the formula:

111 => 11-(1*11) = 11-11 = 0. As a matter of fact, 37 times 3 is 111.

To get a more obvious answer, let us use the square of 37.

1369 => 136-(9*11) = 136-99 = 37.  Yep, 37 is a factor of 37!

One more thing, since 7 is 5+2, if you need to multiply something by 7, you only need to know the twos table and addition.

789 x 7 = ???
789 x 10 = 7890
7890/2 = 3945
789*2  = 1578

                          3945                         

+1578

    5523 

Or, 7 = 10-3.  With 3 = 2+1. Good old twos tables and subtract. Or, of course, just learn the seven's table. Oh, yeah, you can use a calculator!


So, what do I know?

Because of the special relationship between 5 and 2, the tables for the 3 and 7 families look much alike.  Practically mirror images, one finds a "positive" difference by adding, while the other uses the "negative" difference by subtracting.  Each progresses by a factor of 3 between decades.

And so, the chart with only the primes:

07 => 0-(7*2) = 0 -14 =  -14
17 => 1-(7*5) = 1 -35 =  -34
37 => 3-(7*11)= 3 -77 =  -74
47 => 4-(7*14)= 4 -98 =  -94
67 => 6-(7*20)= 6-140 = -134
87 => 8-(7*26)= 8-186 = -174

Prime Factors: Part 3

Prime Numbers:

The Amazing Mr. Five.

Of the prime numbers under 10, only 5 stands alone.  He has no other primes in his family.  As such, recognizing multiples are easy. Just look for a 5 at the end.

This is true, because any number that is divisible by 10 -- that is has a 0 at the end -- divisible by both 2 and 5.  Further determination is made after removing the 0.

For instance:

1234560 is divisible by 2 and 5 (that is, by 10). That removes 5 from the running, at least temporarily.  Here is the progression:

1234560/2*5 = 123456
123456/2    = 61728
61728/2     = 30864
30864/2     = 15432
15432/2     = 7716
7716/2      = 3858
3858/2      = 1929
1929/3      = 543
543/3       = 271

My guess is that 271 is prime. It is one over 270, which has obvious factors.  Further investigation bears me out.

The "Five and Dime" Store.

Being that 5*2 is 10, and inversely 10/2 is 5, multiplying and dividing by 5 is easy. You only need to multiply by 10  and divide by 2 to get any number multiple of 5.

123*5 = 123*10/2 = 1230/2 = 615

Using the decimal point, it works the other way:

(123/2)(10) =61.5*10 = 615

Finally, it is easy to spot numbers divisible by 25 (5^2) and 125 (5^3). This is because 100 and 1000 are multiples of 10.

For 25, there are 3 multiples: 25, 50 and 75.  For example:

123,450 is divisible by 5 and 25. 50 is 2*5*5. So 2, 5, 10 and 25 are factored out.

123,450/2*5 = 12,345
12,345/5    = 6,171
6,171/3     = 2057, an odd number.  Is it prime? Stay tuned.

Another, example:

987,625 is divisible by 125 (5*25=625).  This would also place a 5 on the end of the next factor, raising the power by another 5.

8*125 = 2^3*5^2 = 10^2 = 1000

This leaves 987,000.  Removing the zeros, we then look for the factors, if any, of 987.  3 works, though 9 doesn't.

987/3 = 329.

So, dividing an odd by an odd, we've ended up with an odd number.  Further calculations are needed, but our odd number is within reach.  20*20 is 400, setting a limit.

So, what do I know?

I know that 5 and 10 are closely related by the prime number 2.  And this makes multiplying and dividing by 5 very easy. There are no other prime numbers related to 5, making the possibility of a random number being prime at less than 25%.

Even so, mathematically, there are an infinite number of prime numbers.  This is because "infinity" can not be divided. There are an infinite number of fractions.

Very odd, so to speak, that when not all odd numbers are prime, there can be an infinite number of primes anyway.

I know my brain hurts contemplating that.

To restate the FACTS:

1. The natural number 5 is prime.

2. 5 = 10/2

3. 5*2 = 10

4. Therefore, all numbers ending in 5 or 0 are multiples of 5.

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.   

Making Arrangements

Okay, I've used some of this earlier, but to increase my knowledge, I  have had to "relearn" he technical terms.  While the labels we put on don't really matter, the concepts do.

The most important thing I like to stress about math is knowing how things work.  However, if someone ever asks, there are three "properties" in math that can make things a whole lot easier for those wishing to take control of the numbers they face every day.

Commutative Property

A+B = B+A

A*B = B*A

Yes, this is the very first thing of which I said I am quite certain.  It is one of the undeniable facts of nature.  When it comes to numbers, it makes no difference what order they are in when adding or multiplying.  If you are ever asked what this property is called, just remember the "commute" to work (or school, or wherever).  The numbers are just "moved around."

For this reason, those really scary tables with scores of numbers in them can be reduced in half!  It is a very convenient and time saving fact.  Besides that, it helps you around most multiplication roadblocks.  When memorizing the tables, it is no coincidence that you get the idea that you've seen that "answer" before.  I like to just use the bottom triangular half of the table.

One number says it all on the multiplication table: 49

I almost made that number a separate blog.  It is the square of the number 7, which means it is 7x7.  Anyone who has ever memorized the dreaded "Times Tables" has faced the difficulty of "the sevens."  It is hard to visualize, and difficult to "count by," multiples of 7.  They just about HAVE to be memorized.  That is why the commutative property is so cool.  You don't have to use the "sevens" if you use the OTHER number instead.  Except for one time.  Memorize the fact that 7x7 = 49.

Associative Property

(A+B)+C = A+(B+C)

The associative property is an extension of the commutative property.  It takes advantage of the fact that we can work only with two numbers at a time.  By getting to "round" numbers one can be more confident with the answer.

2+47+64+31+72+99+44+63 = (47+63)+(31+99)+(64+2)+44 
= 110 + 130 + 66 + 44 = 240 + 110 = 350

7 x 9 x 3 x 2 x 5 x 6 = (7x3)x(9x6)x(2x5) = 21 x 54 x 10 = 540 x 21 = ????

So, with multiplication, associative properties only get you so far.  As numbers get larger, another concept is needed to reach the answer. Old fashioned arithmetic has us stack the numbers and then distribute the task using on digit at a time.  That brings us to the next property.

Distributive Property

A(B+C) = AxB + AxC

This property, in which multiplication is spread out over several steps, is the method used practically on scrap paper across the world.  Take the unsolved product of  "540 x 21" for example.

540

x21

The process of distribution is not evident, but it happens as you multiply by 1 and then by 20.  Written in distributive form it looks like this (54)(20 +1). 

(540)(20+1) = (540*20)+(540*1) = 10800 + 540 = 11,340 

Where the distributive property comes in real handy is when calculating products near 10 (8,9,11 and 12).  Though 11 and 12 are derivative of basic facts, they can be bypassed when using a little math. Some math "short cuts" are as follows:

3 = 2 + 1

7 = 5 + 2

8 = 10-2

9 = 10-1

11= 10+1

12 = 10+2

So the take away is that using this property of multiplication, one can "divide and conquer."  One example and then I will let the reader's brain rest.  I will now "randomly" chose two large numbers to multiply.

Let me see, lets do a three digit number by a two digit number, but a little harder than the one above.  I'm randomly picking "794" and "69."

So, a slip of a finger (I meant "8") and blind luck give me 794 x 69.

794 x 69 = (800 - 6)(70 - 1) = [(800x70) -  (6x70)] - 800 -(-6) {I confess, I took a short cut here}

= 56000 - 420 - 800 + 6 = 56,006 - 1220 = 54,786

I really didn't mean to be that complicated, but this is about "what I know."  So using both distributive and associative properties, I took the long way around.  Using rounding, I was able to redistribute the numbers and work inside my head.  

Using the old fashioned arithmetic would be faster, but I showed that I know these three handy properties of numbers!

So, what do I know?

AxB = BxA

A+B = A+B

(A+B)+C = A+(B+C)

A(B+C) = AxB + AxC

Next: Inverse Behavior