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The numbers {a, b,c} are three positive integers - Problem solving

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The numbers {a, b,c} are three positive integers - Problem solving  [#permalink]

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New post 12 Jul 2019, 06:17
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The numbers {a, b,c} are three positive integers. If (a * b * c)/14 equals an integer and (b * c)/4 equals an integer, what is the smallest possible integer value of a?

A. 1
B. 2
C. 4
D. 7
E. 14
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Re: The numbers {a, b,c} are three positive integers - Problem solving  [#permalink]

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New post 12 Jul 2019, 06:23
Least values of a,b,c can be 1,2,7....for a*b*c/14 is an integer and for b*c/4 least values of b,c can be (1,4)(4,1)(2,2)

From above least value of a=7 then b,c can be any of them which satisfies both the equations.....correct me if I'm wron?

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Re: The numbers {a, b,c} are three positive integers - Problem solving  [#permalink]

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New post 12 Jul 2019, 06:45
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madgmat2019 wrote:
Least values of a,b,c can be 1,2,7....for a*b*c/14 is an integer and for b*c/4 least values of b,c can be (1,4)(4,1)(2,2)

From above least value of a=7 then b,c can be any of them which satisfies both the equations.....correct me if I'm wron?

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madgmat2019, I had the same solution as your, but the OA is A. I do not understand the official explanation. Hope someone can help.

--------

TEXT EXPLANATION

This problem is about divisibility. If we want a to be as small as possible, then it would make sense to "take care" of all the divisibility issues with the second fraction, so that there are no requirements that a has to fulfill.

For example, let b = 4 and c = 14, the values of the two denominators. Then

\(\frac{(b * c)}{4}\) = \(\frac{(4 * 14)}{4}\) = 14

So, these choices for b and c satisfy the second condition. Now, look at the first fraction:

\(\frac{(a * b * c)}{14}\)= \(\frac{(a * 4 * 14)}{14}\) = a * 4

The choices we made for b & c allow us to cancel the denominators, so the only requirement now is that a * 4 is an integer. Well, any integer times 4 is still an integer, so that's really no requirement at all. The value a can be any positive integer, so we can make it the smallest positive integer.

a = 1

Answer: (A)
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The numbers {a, b,c} are three positive integers - Problem solving  [#permalink]

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New post 14 Jul 2019, 13:49
The statement does not say the integers but be one digit.

Then a could be 1

B could be 28 and C could be 1 also.

Or

A= 1, B=7, C=4
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Re: The numbers {a, b,c} are three positive integers - Problem solving  [#permalink]

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New post 10 Aug 2019, 11:49
jfranciscocuencag wrote:
The statement does not say the integers but be one digit.

Then a could be 1

B could be 28 and C could be 1 also.

Or

A= 1, B=7, C=4



Actually, it does say that a is an integer...
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Re: The numbers {a, b,c} are three positive integers - Problem solving  [#permalink]

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New post 12 Aug 2019, 09:50
Condition 1 - (a*b*c) is divisible by 14 implies (a*b*c) is divisible by (1*7*2) or (1*1*14)

Condition 2 - (b*c) is divisible by 4 implies (b*c) is divisible by (2*2) or (4*1)

Consider (a*b*c) = 28
Since we have to minimize a, maximize b and c. So (b*c) can be = 28 which satisfy condition 1 and condition 2. Hence a can take the least value which is 1.
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Re: The numbers {a, b,c} are three positive integers - Problem solving   [#permalink] 12 Aug 2019, 09:50
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