## Wednesday, November 10, 2010

### Divisibility and Remainders - If You Imagine It, You Will See It

I am not done with divisibility yet. Not by a long shot. In my last post, I discussed how divisibility was basically making groups and subtracting them out. To understand the basics of what I am going to talk about now, just imagine the numbers as balls. When I say 7, imagine 7 balls in a group.
Let's start easy: Is 7 divisible by 3? Before answering, think of the following figure:
To divide by 3, I have to split 7 into groups of 3. I can make two groups of 3 and then 1 ball is leftover. Hence, when I divide 7 by 3, quotient is 2 and remainder, the ball that could not be put in a group, is 1.

Similarly, what happens when I divide 12 by 3?

I get four groups and nothing leftover. (Is this then the diagrammatic representation of 12 = 3*4? Sure. After all 12/3 is 4, the quotient) Then when I divide 12 by 4, I should be able to get 3 groups with nothing leftover.

I guess you get the picture. Now, a question:
I have a number that when divided by 6, leaves a remainder of 2. What will be the remainder when the number is divided by 3?
So here, I do not know what the number is, but I know that when I make groups of 6, 2 is leftover. Logically, it follows that when I split each of the groups of 6 into two groups of 3 each, I will still have the same remainder of 2.

Then the answer is 2. you don't have to make the diagram in the exam of course! Just imagine it and you will see the answer.

A slight twist on the question above: I have a number that when divided by 6, leaves a remainder of 4. What will be the remainder when the number is divided by 3?
Imagine the picture. Just like above but with groups of 6, there are 4 balls leftover. You divide the groups of 6 into groups of 3, no issues, but now, you can make another group of 3 from the 4 balls that are leftover. Therefore, only one ball will be leftover giving you the remainder of 1. Answer is 1.

Another little twist: A number when divided by 3 gives a remainder of 1. How many distinct values can the remainder take when the same number is divided by 9?
Now imagine that there are lots of groups of 3 and 1 ball leftover. You have to make groups of 9 out of these so you start combining three groups of 3s to make groups of 9. Let's see what the possibilities at the end are:
1. All groups of 3s get used to make groups of 9 and the 1 ball from before is again leftover.
2. One group of 3 is leftover and the 1 ball from before, giving you a total of 4 balls leftover.
3. Two groups of 3 are leftover and the 1 ball from before, giving you a total of 7 balls leftover.
(Three or more groups of 3 cannot be leftover because then, we will be able to make another group of 9 out of them)
Therefore, you can have the remainder in three distinct ways: 1, 4 and 7.
That was easy, wasn't it? I will take more advanced remainder questions in the near future, after discussing factors and multiples.

## Monday, November 8, 2010

### The Scourge and Bane - Exponentials

I am sure many of you hate the sordid looking Indices and Roots. But of course, I don't blame you. Really, who could stand

much less be comfortable solving it. But the sad truth is that Indices and Roots are basics that we should be experts in - whether we are working on Number Systems, Algebra or even Geometry! So, let's get crackin'

Warning: I will just discuss the basic rules in this post for people who dislike exponents.

Remember 'The Four Prongs' post? There we discussed how 34 is just 3 x 3 x 3 x 3. - The '3' is called the base and the '4' is called its index (plural - indices). It follows then that 32 would be just 3 x 3. So now, if I multiply 34 by 32, what do I get? 3 x 3 x 3 x 3 x 3 x 3 = 36 . The indices just got added!

Rule 1: am × an = am + n

Now tell me what happens when I divide
34 by 32 ?

It looks something like the following:

But we know that 3 x 3 =
32 . So essentially, 34 /32 = 34 – 2 = 32

Rule 2:
am / an = am - n
When you divide two numbers with the same base, the index of the divisor gets subtracted from the index of the dividend.

Now that we have discussed Multiplication and Division, we need to think about Addition and Subtraction!
34 to 32 ? What is 34 + 32 ? Can I still add indices? Think. 3 x 3 x 3 x 3 + 3 x 3 = 81 + 9 = 90. I cannot play with the indices when dealing with Addition and Subtraction. The best I can do is take out a common factor. e.g. 34 + 32 = 32 (32 + 1) = 9.10 = 90
I have taken two 3s out from both the numbers and added the rest. Saves me time I need to calculate 3 x 3 x 3 x 3. Exact same thing can be done for Subtraction too.

Now, what is
3-4 ? It is extremely easy to handle negative indices. Just flip the base and the index becomes positive. So 3-4 is 1/34 .
This implies the following:
• 1/3-4 = 34
• 34 = 1/3-4
• 1/34 = 3-4

When you want to change the sign of the index, flip the base. When you want to flip the base, change the sign of the index!

Rule 3: For any number a, a0 = 1
e.g. 30 = 1

How about a quick question now?
Given:

What should replace the ?? to make the equation valid?
I know 4 can be written as 2^2 and I can flip the base in the denominator to the numerator, thereby changing the sign of the index.
We get:

2?? . 22.2-4.2-2 = 1

Here, all the terms are multiplied and there bases are same i.e. 2, so

2??+2-4-2 = 1 = 20

Question: From where did we get 20 ? We know, 20 = 1 so if we have 1, we can write it as 20.

That is the only way in which a term with base 2 could have become 1.

Now, since the bases on both sides of the equation are same, the

indices should also be the same.

?? + 2 - 4 - 2 = 0

?? = 4

Note: Of course, you could have cancelled off 4 in the numerator with 22 in the denominator and got your answer directly, but here we are learning to use the various rules of exponents.