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If a, b, c and d positive integers such that they satisfy the system o

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nick1816
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If a, b, c and d positive integers such that they satisfy the system o [#permalink] New post   03 Aug 2020, 18:47
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If a, b, c and d positive integers such that they satisfy the system of equations above, find a+b+c+d.

A. 12
B. 14
C. 16
D. 18
E. 21
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nick1816
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Re: If a, b, c and d positive integers such that they satisfy the system o [#permalink] New post   04 Aug 2020, 12:10
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Another way is

ac+bd = 34
ad+bc=43

Add both equations

ac+ad+bd+bc = 77

a(c+d)+b(c+d) = 77

(a+b)(c+d) = 1*77 or 7*11

1*77 case is not possible, since all of them are positive integers.

(a+b) and (c+d) are equal to 7 and 11 in any order.

a+b+c+d = 7+11=18
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Re: If a, b, c and d positive integers such that they satisfy the system o [#permalink] New post   04 Aug 2020, 09:59
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Re: If a, b, c and d positive integers such that they satisfy the system o [#permalink] New post   04 Aug 2020, 14:48
nick1816 wrote:
Another way is

ac+bd = 34
ad+bc=43

Add both equations

ac+ad+bd+bc = 77

a(c+d)+b(c+d) = 77

(a+b)(c+d) = 1*77 or 7*11

1*77 case is not possible, since all of them are positive integers.

(a+b) and (c+d) are equal to 7 and 11 in any order.

a+b+c+d = 7+11=18


nick1816 does it mean 1st equation i.e. ad+cd=38 makes no sense.
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Re: If a, b, c and d positive integers such that they satisfy the system o [#permalink] New post   04 Aug 2020, 15:08
There are multiple ways to find the value of a+b+c+d. There are solutions in which all 3 equations are required (if i find the values of a, b, c and d individually and then add them) . But the way i solved this problem, equation 1 is certainly redundant.

ammuseeru wrote:
nick1816 wrote:
Another way is

ac+bd = 34
ad+bc=43

Add both equations

ac+ad+bd+bc = 77

a(c+d)+b(c+d) = 77

(a+b)(c+d) = 1*77 or 7*11

1*77 case is not possible, since all of them are positive integers.

(a+b) and (c+d) are equal to 7 and 11 in any order.

a+b+c+d = 7+11=18


nick1816 does it mean 1st equation i.e. ad+cd=38 makes no sense.
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If a, b, c and d positive integers such that they satisfy the system o [#permalink] New post   04 Aug 2020, 15:13
nick1816 ,
Alright, can I ask what first thing did come in your mind when you saw this question ? First key point!
Because when I saw this question even I starting adding all equations and tried to find a+b+c+d.

If possible, can you point me similar questions. I just wanna make sure when I see such question in exam, I get ur approach.
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Re: If a, b, c and d positive integers such that they satisfy the system o [#permalink] New post   02 Nov 2022, 05:24
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Re: If a, b, c and d positive integers such that they satisfy the system o [#permalink]
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