Fundamental Arithmetic Theorem

Fundamental Arithmetic Theorem

Step 1

Ask students if 4 can be divided with a number not 1 or itself i.e 4 (because all numbers can be divided with 1 and itself)
They will answer: YES with 2
Now ask them this question about 5. Result NO
Now ask them this question about 9. Result YES
Now ask them this question about 13. Result NO
Now ask them to find out some more examples and list it which are divided and which are not divided.
THOSE NUMBERS CAN NOT BE DIVIDED AGAIN ARE CALLED PRIME FACTORS

Step 2

Now ask them to express 36 as multiple of numbers
They should do                                                          6 X 6
Ask them again to express each factor in the result to be multiple of two numbers
They should do                                                          3 X 2 X 6 Again 3 X 2 X 3 X 2
Now it cannot be done any more.
So 36 can be expressed as 3 X 2 X 3 X 2

Ask them to do this exercise with 92, 39 and 72.

So, any number can be expressed as product of prime factors.

Step 3

Again. back to 36. It's expressed as 3 X 2 X 3 X 2 i.e. product of two 2s and two 3s.
Ask them what number should get as a product of two 2s and two 3s?
It's only 36, not any else.
For any number not 36 there should be some change in its prime factorization (i.e. this model expression).
So every number have separate group of prime factors i.e. UNIQUE.

Step 4

3 X 2 X 3 X 2 is also equal to 2 X 2 X 3 X 3
So, there is no change if the order is not considered.

Theory:

Every number can be expressed as product of UNIQUE PRIME FACTORS, irrespective of the order.