OG: Factorization

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OG: Factorization

by zacharyz » Sat May 17, 2008 4:00 pm
From OG:

If r and s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r
(2) Every prime factor of s is also a prime factor of r

OA is A






I understand the answer for the most part. My question is if there is standard wording to differentiate between simply sharing prime factors and sharing exactly the right number of prime factors for (2).

Ex. if r is 18 and s is 8. In this case, the prime factors of r are 2*3*3. The prime factorization of s is 2*2*2. However the argument is that the wording really specifies that the prime factor of 8 is 2 and the prime factor of 18 is 2 and 3, but 18/8 is not an integer. What would it say to be clear that it did mean the entire prime factorization (meaning that if 8 was s, then r would have to have three 2's in its factorization)?

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by luvaduva » Sat May 17, 2008 11:33 pm
This is how I worked the problem when I first saw it.

(1) Infers that r > s or r = s.

If r =12 and s =6
Factors of 12 = 1, 2, 3, 4, 6, and 12
Factors of 6 = 1, 2, 3, and 6
12/6 = 2 => Integer

If r = 12 and s =12
12/12 = 1 => Integer

Sufficient

(2) No inference to whether r > s, r = s, or s > r

If r = 12 and s = 6
Prime Factorization of 12 = 2*2*3
Prime Factorization of 6 = 2*3

The prime factors of 12 are 2 and 3. The prime factors of 6 are 2 and 3.

12/6 = 2 => integer

If r = 6 and s =12
6/12 = .5 => not an integer

Not sufficient

That was my first reaction to the problem, but looking back at it the problem IS a bit ambiguous.