414 = 2 X 207 = 2 X 9 X 23 = 2 X 3 X 3 X 23.
So the prime factorization of 414 is 2 X 32 X 23.
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No, 3x5x4 is not a prime factorization of 60. A prime factorization of 60 would consist of only prime numbers. The prime factorization of 60 is 2x2x3x5.
When all the factors are prime numbers, that's a prime factorization.
Prime factorization!
There is no need to do prime factorization as prime numbers are already prime.
683 is already prime; no factorization.