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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, 05:17
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A
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Difficulty:

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Question Stats:

52% (01:33) correct 48% (01:11) wrong based on 108 sessions

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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, 05:45
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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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Re: The numbers {a, b,c} are three positive integers - Problem solving  [#permalink]

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New post 12 Jul 2019, 05:23
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?

Posted from my mobile device
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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, 12:49
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, 10:49
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, 08:50
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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The numbers {a, b,c} are three positive integers - Problem solving  [#permalink]

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New post 07 May 2020, 07:30
Starfruit wrote:
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



SOLUTION:
Pick the first monomial and multiply by 2/2 (which is equal 1), split the denominator in 7*2, then simplify and divide the two terms of the product:
(a * b * c)/14=(2a/7)*[(b * c)/4]=(2a/7)*[(b * c)/4]
Remarks:
- [(b * c)/4] is an integer (we don't know if odd or even)
- (2a/7) must be an integer!! (if the blue term were even this could have also been values like 0.5, 1.5, 2.5, 3.5, etc... but we're not in that case)

WHY IS THE TEXT EXPLANATION WRONG?
If a=1 we get:
- (2/7)*[(b * c)/4]=[(2 * b * c)/(4 * 7)]=integer -----> true only if (b*c) is divisible by 2 (it's given that is divisible by 4) and 7 -not given- (like in the particular case used picking numbers by the text, b=4 and c=14... indeed taking a different "c" -not multiple of 7- would lead to a totally different result)


Among the options the smallest possible integer value for "a" that guarantees (2a/7) to be an integer is clearly:
a=7


CORRECT ANSWER: D
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The numbers {a, b,c} are three positive integers - Problem solving   [#permalink] 07 May 2020, 07:30

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

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