If I am given all the prime factors of an integer, does that mean I know the value of that number?
For example, if all the prime factors of a certain number are 1, 2, and 3, the number could be 6 or 12, right?
I ask because the answer explanation for OG 13, Diagnostic # 26 (page 25) states that if I am given all the prime factors of an integer, I would know the value of that number by multiplying all the factors together. I don't know why that is a true statement.
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- QuinganZ341
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- RonPurewal
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Re: prime factors of a number
QuinganZ341 Wrote:the answer explanation for OG 13, Diagnostic # 26 (page 25) states that if I am given all the prime factors of an integer, I would know the value of that number by multiplying all the factors together.
the explanation states no such thing.
- RonPurewal
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Re: prime factors of a number
ok, i see where you're going with this. there are two possible objections: (a) there could be other factors, and/or (b) the given factors could be repeated.
the explanation chose to mention (a). since (a) is already enough to prove that the answer is E, there's really no need to mention (b) one way or the other.
from the fact that (b) isn't mentioned one way or the other (because there's no need), you're mistakenly inferring that the explanation is dismissing that possibility. nope.
analogy:
Ron, can your niece be elected president?
-- Nope. She's not 35 years old yet. (the minimum age for US presidents, per the constitution, is 35 years)
^^ that's all i have to say to prove that she can't be elected president; there's no need to go into other reasons. (with your logic above, you would mistakenly infer that ANY 35+-year-old could be elected president.)
the explanation chose to mention (a). since (a) is already enough to prove that the answer is E, there's really no need to mention (b) one way or the other.
from the fact that (b) isn't mentioned one way or the other (because there's no need), you're mistakenly inferring that the explanation is dismissing that possibility. nope.
analogy:
Ron, can your niece be elected president?
-- Nope. She's not 35 years old yet. (the minimum age for US presidents, per the constitution, is 35 years)
^^ that's all i have to say to prove that she can't be elected president; there's no need to go into other reasons. (with your logic above, you would mistakenly infer that ANY 35+-year-old could be elected president.)
- RonPurewal
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Re: prime factors of a number
...and, finally, 1 is not a prime number.
this is not random, by the way; it's essential to exclude 1, in order for primes to do what they do.
perhaps the most important thing about primes is that prime factorizations are unique. e.g., 10 factors into 2 x 5, and that's the only way to factor it into primes.
if we were to allow 1 as a prime number, that uniqueness would be destroyed, since you could then say "ha! it's also 2 x 5 x 1, or 2 x 5 x 1 x 1, or ..."
so, yeah, 1 isn't prime.
this is not random, by the way; it's essential to exclude 1, in order for primes to do what they do.
perhaps the most important thing about primes is that prime factorizations are unique. e.g., 10 factors into 2 x 5, and that's the only way to factor it into primes.
if we were to allow 1 as a prime number, that uniqueness would be destroyed, since you could then say "ha! it's also 2 x 5 x 1, or 2 x 5 x 1 x 1, or ..."
so, yeah, 1 isn't prime.
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