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If a,b and c are all positive integers satisfying a*b*c= 7(a+b+c), wha

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nick1816
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If a,b and c are all positive integers satisfying a*b*c= 7(a+b+c), wha [#permalink] New post  10 Apr 2020, 00:03
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If a,b and c are all positive integers satisfying \(a*b*c= 7(a+b+c)\), what is the smallest possible value of \((a+b+c)\)?

A. 12
B. 15
C. 18
D. 21
E. 24
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preetamsaha
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Re: If a,b and c are all positive integers satisfying a*b*c= 7(a+b+c), wha [#permalink] New post  10 Apr 2020, 00:29
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If a,b and c are all positive integers satisfying a∗b∗c=7(a+b+c), what is the smallest possible value of (a+b+c)?

here, a∗b∗c=7(a+b+c)
already one factor 7 . let , a=7
so, (a+b+c) must be a number of two factors so that b,c can be positive integers but keeping one condition in mind that is,
(a+b+c) = bc

question wants to find smallest number .so starting from choice a,
(a+b+c) = 12 =2*6
so, b=2, c=6
and a= 7 but (a+b+c) = 7+2+6 = 15 not 12 (rejected )
(a+b+c) = 12 = 3*4
so, b=3, c=4
and a= 7 but (a+b+c) = 7+3+4 = 14 not 12 (rejected )

(a+b+c) = 15 =5*3
so, b=5, c=3
and a= 7 but (a+b+c) = 7+5+3 = 15 not 15 (accepted )

correct answer B
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Re: If a,b and c are all positive integers satisfying a*b*c= 7(a+b+c), wha [#permalink] New post  09 Oct 2020, 22:09
Is there any other way besides brute force method for this problem?
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Re: If a,b and c are all positive integers satisfying a*b*c= 7(a+b+c), wha [#permalink] New post  09 Oct 2020, 23:52
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a*b*c= 7(a+b+c)

So one of a,b and c must be a multiple of 7. Since we have to minimize a+b+c, one of them is equal to 7(minimum positive multple of 7).

Consider c=7

a*b=a+b+7

a*b-a-b= 7

a*b-a-b+1 = 7+1

a(b-1)-1(b-1)=8

(a-1)(b-1) = 1*8 or 2*4

a=2 and b=9 or a=3 and b=5

To minimize a+b+c, consider 2nd case; a=3 and b=5.


KeyurJoshi wrote:
Is there any other way besides brute force method for this problem?
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Re: If a,b and c are all positive integers satisfying a*b*c= 7(a+b+c), wha [#permalink] New post  16 Aug 2021, 02:16
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a*b*c=7(a+b+c) in this substitute option, will get easily answer.
Lets choose option A=12
7(a+b+c)=7*12=84 make a factors of 84 =4*3*7
it's not equal to a+b+c=12

Now choose option B=15
7(a+b+c)=7*15=105 make a factors of 105
105=3*5*7
a*b*c=7(a+b+c)
3*5*7=7(3+5+7)

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Re: If a,b and c are all positive integers satisfying a*b*c= 7(a+b+c), wha [#permalink] New post  07 Oct 2022, 00:55
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Re: If a,b and c are all positive integers satisfying a*b*c= 7(a+b+c), wha [#permalink]
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