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bekerman
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Factors and Prime Factors 30 Jan 2014, 10:34
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I just wanted to confirm my understanding of a topic as I think I might have seen differing definitions for this subject before. (A credible source would help me here as well.)

So, the FACTORS of the number 189 are 1, 3, 7, 9, 21, 27, 63, 189, which does NOT take into account that 189's prime factorization includes multiple 3's.

The PRIME FACTORS of the number 189 are a subset of its factors, which would be 3 and 7, which does not take into account repetition of 3's, right?

By these definitions, is the term 'DIFFERENT' prime factors essentially repetitive? 2*2*3*2 would have 2 prime factors and so would 2*3, yes?



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Factors and Prime Factors 30 Jan 2014, 11:54
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bekerman
I just wanted to confirm my understanding of a topic as I think I might have seen differing definitions for this subject before. (A credible source would help me here as well.)

So, the FACTORS of the number 189 are 1, 3, 7, 9, 21, 27, 63, 189, which does NOT take into account that 189's prime factorization includes multiple 3's.

The PRIME FACTORS of the number 189 are a subset of its factors, which would be 3 and 7, which does not take into account repetition of 3's, right?

By these definitions, is the term 'DIFFERENT' prime factors essentially repetitive? 2*2*3*2 would have 2 prime factors and so would 2*3, yes?
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Dear bekerman,
I'm happy to help. :-)

First of all, here's a post you may find helpful.
https://magoosh.com/gmat/2012/gmat-math-factors/

The factors of 189 are, as you say, {1, 3, 7, 9, 21, 27, 63, 189}. Eight total factors, including 1 (the universal factor) and the number 189 itself.

The prime factorization of 189 is 189 = 3*3*3*7 = (3^3)*7

The question "how many prime factors does 189 have?" is potentially ambiguous --- it has four total prime factors, but only two are distinct. (Mathematicians use the word "distinct" to mean "different.") The GMAT itself will not ask that question.

The question, "how many distinct prime factors does 189 have?" is a clear question with a clear answer: 189 has two distinct prime factors, 3 and 7. The numbers 21, 63, 147, etc. all have the same two distinct prime factors that 189 has.

The prime factorization of each positive integer greater than 1 is unique. It's like the DNA of the number --- if you know N's prime factorization, you know all the potential factors of N. The list of distinct prime factors of a number is not unique --- as we saw, 21, 63, 147, 189, and many more have the exact same list of distinct prime factors. Knowing that N has two distinct prime factors, 3 and 5, does not tell us what N is, but it does let us know a great deal about divisibility and related properties (N must be odd, N is not divisible by 7 or 11, etc. )

Does all this make sense?
Mike :-)



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