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If a and b are positive integers such that a/b = 2.86, which of the following must be a divisor of a?

1. 10 2. 13 3. 18 4. 26 5. 50

\(\frac{a}{b}=2.86=\frac{286}{100}=\frac{143}{50}\) --> \(b=\frac{50a}{143}=\frac{50a}{11*13}\), for \(b\) to be an integer \(a\) must have all the factors of 143 (50 has none of them). Hence \(a\) must be divisible by both 11 and 13.

If a and b are positive integers such that a/b = 2.86, which of the following must be a divisor of a?

1. 10 2. 13 3. 18 4. 26 5. 50

\(\frac{a}{b}=2.86=\frac{286}{100}=\frac{143}{50}\) --> \(b=\frac{50a}{143}=\frac{50a}{11*13}\), for \(b\) to be an integer \(a\) must have all the factors of 143 (50 has none of them). Hence \(a\) must be divisible by both 11 and 13.

Answer: B.

Hi Bunuel,

I'm trying to follow MGMAT's method: we know that a/b = 2.86 2.86 => 2 and 86/100 or 43/50 and we know that r/b = 43/50 hence 50r = 43b from that we conclude that b must be a multiple of 50 and 43 a multiple of r. What am I doing wrong?

Re: If a and b are positive integers such that a/b = 2.86, which [#permalink]
13 Apr 2015, 09:34

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Re: If a and b are positive integers such that a/b = 2.86, which [#permalink]
16 Apr 2015, 19:09

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Expert's post

Hi All,

The prompt gives us a couple of facts to work with: 1) A and B are positive INTEGERS 2) A/B = 2.86

We can use these facts to figure out POSSIBLE values of A and B. The prompt asks us for what MUST be a divisor of A. Since we're dealing with a fraction, A and B could be an infinite number of different integers, so we have to make both as SMALL as possible; in doing so, we'll be able to find the divisors that ALWAYS divide in (and eliminate the divisors that only SOMETIMES divide in).

The simplest place to start is with... A = 286 B = 100 286/100 = 2.86

These values are NOT the smallest possible values though (since they're both even, we can divide both by 2)...

A = 143 B = 50 143/50 = 2.86

There is no other way to reduce this fraction, so A must be a multiple of 143 and B must be an equivalent multiple of 50. At this point though, the value of B is irrelevant to the question. We're asked for what MUST divide into A....

Since A is a multiple of 143, we have to 'factor-down' 143. This gives us (11)(13). So BOTH of those integers MUST be factors of A. You'll find the match in the answer choices.

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