- Level: International Baccalaureate
- Subject: Maths
- Word count: 2295
Math Portfolio higher level type 1
Extracts from this document...
Introduction
Investigating Divisibility
In order to determine if an expression is divisible by a certain value, we factorize the expression and see if we can take the corresponding value let's call it as a common factor. Afterwards, we see if it is divisible depending on how the expression will turn out. I'll explain more with examples.
Now let's look at the expression. Now we want to see if the expression is always divisible by the corresponding.
The first case if =2.
Now by substituting 2 in the expression, the expression will look like this:
Now let's take as a common factor. The expression will become and since the expression or it is therefore divisible by 2.
Now let's check the validity of my statement let's take a few examples.
Let
Using GDC we plug the following values in the expression and check if it is divisible by 2.
And 20 is divisible by 2.
Again 182 is divisible by 2.
Also 280 is divisible by 2
Which is divisible by 2
Therefore is divisible by 2.
Now let's take the second case when
Now by substituting 3 in the expression it will turn out to be like this:
Let's take as a common factor the expression will now look like this:
And now by factorizing it more where is difference between two squares, the expression will look like this which is three successive (consecutive) terms.
Therefore, is divisible by 3.
To make sure this is true let's take a few examples.
Let
Now using GDC substitute the following values of in the expression
Which is divisible by 3
And this number is divisible by 3
Middle
However, we need to prove that is true.
And .
Now let's subtract from.
The expression is now
Let's solve the brackets so the expression becomes in this form:
By collecting terms and simplifying the
Expression it will become:
And by taking as common factor the expression is now:
Therefore it is divisible by 3
Since is not divisible by 4 it is ignored and we don't have to prove by induction.
The 3rd case is when
Now by using mathematical induction we want to prove that is divisible by 5.
First we let n=1 and see if it is divisible by 5
Therefore divisible by 5.
Now we assume that is true so is true.
However, we need to prove that is true.
And .
Now let's subtract from.
The expression is now
Let's solve the brackets, so the expression becomes in this form:
By collecting terms and simplifying the
Expression it will become:
And by taking as common factor the expression is now:
Therefore it is divisible by 5.
Now let's explore more cases for and if is divisible bywe'll prove it by induction. So we will factorize the expression for
Let's look at when
Now let's plug the value of in the expression.
Now let's keep factorizing the expression more
I don't find any clear evidence in the expression to show if the expression is divisible by 6 or not however, let's take a few example to check.
Let
Now by plugging the following values in the expression we check if it is divisible by 6.
which is not divisible by 6
Which is not divisible by 6.
Conclusion
After finding this relationship I came up with a new conjecture. If in the expression is a prime number then is divisible by . Also, if is a prime number in Pascal's triangle where is the number of rows then the entries in the row are divisible by and it is a multiple of.
However, if we look at the converse of this statement which is if expression divides by then is a prime number. This statement is not true since this expression can divide by some non-prime numbers for some values of.
For example let's take 4, although 4 is not prime yet if you plug 5 in the expression instead of it will divide by 4.
which is divisible by 4.
Therefore, the converse of my conjecture is not true and doesn't hold.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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