If a number is a product of unique prime numbers, are the factors of this number the used unique prime numbers ONLY? Example: 6 = 2 x 3, 15 = 3 x 5. But I don't know for large numbers. I will be using this in my code to speed up my checking on uniqueness of data. Thanks! 😀
Edit:
I will be considering all unique PRIME factors only. For example, I will not generate 9 because it's factors are both 3 (I don't consider 1 here), And also 24 (= 2 x 2 x 2 x 3). I want to know if it is TRUE if unique PRIME numbers are multiplied, the product's PRIME factors are only those PRIME factors that we multiplied in the first place. Sorry for not clarifying it earlier.
Best Answer
If I understand you correctly (which I realize I didn't), the answer is no. Take the number $30=2\cdot 3 \cdot 5$ for example. $6$, $10$, and $15$ are also factors of $30$. $$30 = 2 \cdot 3 \cdot 5 = (2 \cdot 3) \cdot 5 = 6 \cdot 5 = 2 \cdot (3 \cdot 5) = 2 \cdot 15 = (2\cdot 5) \cdot 3 = 10 \cdot 3$$
Edit: After your clarification, I will change my answer to yes. See the fundamental theorem of arithmetic. It "states that any integer greater than 1 can be written as a unique product (up to ordering of the factors) of prime numbers." - Wikipedia.