I have a conceptual question:
Does each positive integer have a unique combination of prime factors when we make its prime factorization?
In other words, when we calculate the prime factorization of a number, will we get always the same answer (combination)?, or there could be different sets of prime numbers (including repetitions) whose product is the same number?
For example, 100 = \(2^2\)*\(5^2\) , in this case is the only set of prime factors whose product is 100. However, I wonder whether in other numbers the opposite is possible.
Please, provide a detailed explanation.
A lot of further concepts depend on a thorough understanding on this.
Think of prime factors as basic indivisible building blocks for numbers.
2 - Red block
3 - Yellow block
5 - Green block
and so on..
To make 4, you need 2 red blocks.
To make 100, you need 2 red and 2 green blocks
To make 300, you need 2 red, 2 green and 1 yellow block.
Conversely, whenever you take 1 red and 1 yellow block, you will get 6.
Whenever you take 1 yellow block and 1 green block, you will get 15.
and so on...
Whenever you take 2 red and 2 green blocks, you will always get a 100. Whenever you try to break down 100, you will always get 2 red and 2 green blocks.
You can write 100 as 10*10 but each 10 is made up of 1 red and 1 green block so finally you have 2 red and 2 green blocks only.
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