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# The numbers {a,b,c} are three positive integers.

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The numbers {a,b,c} are three positive integers. [#permalink]  03 Sep 2019, 03:08
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The numbers {a,b,c} are three positive integers. If $$\frac{(a×b×c)}{14}$$ equals an integer and $$\frac{(b×c)}{4}$$ equals an integer, what is the smallest possible integer value of a?

A. $$1$$
B. $$2$$
C. $$4$$
D. $$7$$
E. $$14$$
[Reveal] Spoiler: OA

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Re: The numbers {a,b,c} are three positive integers. [#permalink]  03 Sep 2019, 08:09
Expert's post
Setting the two conditions $$\frac{a \times b \times c}{14}$$ AND $$\frac{b \times c}{14}$$ we do know that $$b\times c$$ must be a multiple of 14: 14,28, so forth.

The question boils down to the value of a in the fraction above. To continue to have an integer a must be equal to 1 as minimum value to get the overall fraction an integer

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Re: The numbers {a,b,c} are three positive integers. [#permalink]  03 Sep 2019, 21:00
Carcass wrote:
Setting the two conditions $$\frac{a \times b \times c}{14}$$ AND $$\frac{b \times c}{14}$$ we do know that b\times c must be a multiple of 14: 14,28, so forth.

The question boils down to the value of a in the fraction above. To continue to have an integer a must be equal to 1 as minimum value to get the overall fraction an integer

Regards

still confuse
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Re: The numbers {a,b,c} are three positive integers. [#permalink]  04 Sep 2019, 02:46
Expert's post
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Re: The numbers {a,b,c} are three positive integers. [#permalink]  04 Sep 2019, 04:26
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Re: The numbers {a,b,c} are three positive integers. [#permalink]  05 Sep 2019, 09:52
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huda wrote:
The numbers {a,b,c} are three positive integers. If $$\frac{(a×b×c)}{14}$$ equals an integer and $$\frac{(b×c)}{4}$$ equals an integer, what is the smallest possible integer value of a?

A. $$1$$
B. $$2$$
C. $$4$$
D. $$7$$
E. $$14$$

Notice that 28 is divisible by both 14 and 4.
So, if b equals 28, then abc/14 and bc/4 are both integers, REGARDLESS of the values of the other two variables.

For example, if a = 1, b = 28 and c = 1, then abc/14 and bc/4 both simplify to be integers.

Since variable a must be a POSITIVE INTEGER, the smallest possible value of a = 1

Cheers,
Brent
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Re: The numbers {a,b,c} are three positive integers. [#permalink]  05 Sep 2019, 11:02
Ok great this is nice thank you
Re: The numbers {a,b,c} are three positive integers.   [#permalink] 05 Sep 2019, 11:02
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# The numbers {a,b,c} are three positive integers.

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