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Re: If a and b are positive integers, is a a multiple of b? [#permalink]
chetan2u

how come for statement 2 you have not taken case of p>r or q>s
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Re: If a and b are positive integers, is a a multiple of b? [#permalink]
Expert Reply
energetics wrote:
If a and b are positive integers, is a a multiple of b?

(1) Every distinct prime factor of b is also a distinct prime factor of a.

(2) Every factor of b is also a factor of a.

Question: is a = bk+0/a? In other words, can we divide a by b with 0 remainder?

1) Take 2,3,5 as the prime factorization.
2*3*5/2*3*5 = 1 so yes, a/b ONLY if they have the same amount of each prime, but we don't know this.
If we change the power of the primes in b, (e.g. 2*3*5/2^2*3*5) then it's no since it's a non-int.
Not sufficient.

2) If every factor of b is in a, then a is a multiple of b.
Take 2,3,5 again as the prime factorization. Factors are 1*30, 2*15, 3*10, 5*6. Any way we choose we get a=30, b=30, 30/30 = 1, so Yes.
Sufficient.

Official explanation:
To solve this yes-or-no problem, your best bet is to plug in values that meet the requirements given in the statements to determine whether Plugging In always yields the same answer to the question.

For Statement (1), for example, we could Plug In 4 for both a and b—that’s one easy way to make sure that every prime factor of b is also a prime factor of a—and since every number is a multiple of itself, the answer to the question is “yes.” However, if we leave b = 4 but make a = 2, we can still satisfy the requirement of Statement (1)—4 has only one prime factor, 2, which is also a prime factor of 2—but now our answer is “no.” Since Statement (1) yields different answers, it’s insufficient, and the possible
answers are BCE.

Similar attempts in Statement (2), however, will always yield the answer “yes.” We could, of course, again Plug In 4 for both variables, and we have our first “yes.” If we leave b = 4, though, we can’t make a = 2, since 4 isn’t a factor of 2; a is a number such as 4 (which we’ve already used), 12, 16, or some other number that has 1, 2, and 4 as factors. Whichever one we pick, though, our answer is “yes”; Statement (2) is therefore sufficient, and the answer to the problem is (B).


Solution:
Pre Analysis:
  • a and b are positive integers
  • We are asked if a is a multiple of b or not
  • This is a yes-no question

Statement 1: Every distinct prime factor of b is also a distinct prime factor of a
  • Let us assume \(b=2^3\times 3^2\)
  • Now if \(a=2^3\times 3^3\) then a is a multiple of b
  • However, if \(a=2^3\times 3\) then a is not a multiple of b
  • Thus, statement 1 alone is not sufficient and we can eliminate options A and D

Statement 2: Every factor of b is also a factor of a
  • According to this statement, a might or might not have any extra factor apart from all the factors of b
  • However, in both cases, we can be sure that a is a multiple of b
  • Thus, statement 2 alone is sufficient

Hence the right answer is Option B
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Re: If a and b are positive integers, is a a multiple of b? [#permalink]
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Re: If a and b are positive integers, is a a multiple of b? [#permalink]
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