First, make sure all the factors are, in fact, prime. Then make sure that the product of all the factors is the number you wanted to factor.
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60 30,2 15,2,2 5,3,2,2 2 x 2 x 3 x 5 is the correct prime factorization for 60 because they're all prime numbers and their total product is 60.
That is correct, the prime factorization of 240 is 24 x 3 x 5
175/245 = 35/49 = 5/7 these are both prime numbers
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.