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This came up in my review, my answer was incorrect.
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.
Can you provide rationale as well?
E - each statement is sufficient. here is my rational:
1 - if each factor of r is also a factor of s then when you break down the numbers into their factors you will cancel each other out and have an integer. actually this will probably yield to one. 2 - each number can be broken down to a set of prime factors. a number is either prime(divisible only by 1 and itself) or can be broken down into prime numbers. so the same thing here as with 1. all prime will cancel each other out and yield an integer.
(2) Every prime factor of s is also a prime factor of r.
2 is insufficient because consider s=4 r=10
prime factor of S is 2 prime factor of r is 5 and 2
so back to stat 2, every prime factor of S (which 2 is the only prime factor) is also a prime factor of R (which is 10) well 2 is prime factor of 10 as well
First statement sufficient as every factor of s is a factor of r as well so they cancel out to make an integer Remember we cannot take example of r=6 and s=4 as somebody has taken previously as there are two factors of two in 4 but only one factor of 2 in 6. Statement 2 insufficient e.g r=9 and s=6 and r=9 and s=3
I'm not a fan of this question. If, for example S = 12, its prime factorization is 2^2 * 3.
According to statement 2, EVERY prime factor of s (I took this to mean 2, 2, and 3 since it has two 2's, which are both prime factors). Basically I thought the trick was that you had to understand that for every integer, there is a prime factorization, and therefore the two statements were basically the same. I guess I misinterpreted (2), but still, how would you KNOW how to interpret that?
I'm not a fan of this question. If, for example S = 12, its prime factorization is 2^2 * 3.
According to statement 2, EVERY prime factor of s (I took this to mean 2, 2, and 3 since it has two 2's, which are both prime factors). Basically I thought the trick was that you had to understand that for every integer, there is a prime factorization, and therefore the two statements were basically the same. I guess I misinterpreted (2), but still, how would you KNOW how to interpret that?
What a worst coincidence! I did the same. I was looking for numbers that had the same prime factors. But I guess we are wrong.
Every prime factor of s is also a prime factor of r.
does not mean that every prime factor of s need not be a prime factor of r as many times in s. It can just be a subset of the numbers. as some said 4,8 good 4,6 not good