Sunday, December 30, 2007

Algebra - Numbers Divisible by 9 (Friendly Version)

How do you know when a number is divisible by 9? If the sum of its digits is divisible by 9, then the original number is divisible by 9! For example, 6525 is divisible by 9 because the sum of its digits (6 + 5 + 2 + 5 = 18) is divisible by 9 (18/9 = 2). But, how and why does this happen?9 times 1 is 9, and 9 is divisible by 9; 9 times 2 is 18, and 18 is divisible by 9; 9 times 3 is 27 and 27 is divisible by 9; so, in general, 9 times any number is divisible by 9. 9x is divisible by 9; 9y is divisible by 9. It doesn't matter; x can be a number or a long sum of numbers; just as long as you multiply by 9, you can always divide a 9 out. Let us remember 9x. We want to get a number in the form of 9x. Then we know it has to be divisible by 9.Take 6525 for example! Now ask yourself if the mathematical sentences are true or not.
Is sentence below true?
6525 = 6(1000) + 5(100) + 2(10) + 5(1)
Yes; look at the metric system: 6m + 5dm + 2cm + 5mm = 6525mm
Is sentence below true?
6525 = 6(999 + 1) + 5(99 + 1) + 2(9 + 1) + 5
Again, yes because 10 = 9 + 1; 100 = 99 + 1, and so on.

Is the sentence below true?
6525 = 6(999) + 5(99) + 2(9) + 6 + 5 + 2 + 5
Yes, because we are allowed to distribute numbers like this. For example, x(x + 2) = x^2 + 2x, or 25 = 5(5) = 5(2 + 3) = 5(2) + 5(3) = 10 + 15 = 25. Now notice that the numbers on the right are the sum of the number 6525, which is 6 + 5 + 2 + 5 = 18, and 18 is divisible by 9 because 9 times 2 =18, and again 9 times any number is divisible by 9.

Is this sentence below true?
6525 = 9(111)(6) + 9(11)(5) + 9(2) + 9(2)
Yes, because all we did was factor out a 9 from each term. The last term was the sum of the digits 6525.This sentence below is true because we can factor a 9 from each term, we couldnt have done this if the sum of the digits in 6525 was NOT divisible by 9, so we need the sum of the digits to equal a number that's divisible by 9 in order to factor it out at this step!
6525 = 9[6(111) + 5(11) + 2(1) + 2]

Now remember, we said 9 times ANY NUMBER is divisible by 9, well let's make that number [6(111) + 5(11) + 2(1) + 2], now we have shown not only how but why if the sum of digits are equal to a number divisible by 9, that number MUST be divisible by 9, and vis versa. Here are all the steps in order:
6525 = 6(1000) + 5(100) + 2(10) + 5(1)6525 = 6(999 + 1) + 5(99 + 1) + 2(9 + 1) + 5
6525 = 6(999) + 5(99) + 2(9) + 6 + 5 + 2 + 5
6525 = 9(111)(6) + 9(11)(5) + 9(2) + 9(2)
6525 = 9[6(111) + 5(11) + 2(1) + 2]
6525/9 = 6(111) + 5(11) + 2(1) + 2725 = 666 + 55 + 2 + 2 = 725

1 comment:

Meryl Streep Rocks said...

Hey, my teacher taught me that stuff as well. It's not like NO ONE knows how to do it!
Lachie:-)