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If for integers a, b, c, and d, a + b + c + d = 4, does d = 1?

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If for integers a, b, c, and d, a + b + c + d = 4, does d = 1? [#permalink]

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New post 21 Oct 2017, 00:53
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Question Stats:

26% (00:57) correct 74% (01:09) wrong based on 31 sessions

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If for integers a, b, c, and d, a + b + c + d = 4, does d = 1?

(1) abc = 1
(2) ad = 1
[Reveal] Spoiler: OA

Last edited by Bunuel on 21 Oct 2017, 01:02, edited 1 time in total.
Renamed the topic, edited the question and added the OA.

Kudos [?]: 18 [0], given: 158

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Kudos [?]: 256 [1], given: 34

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If for integers a, b, c, and d, a + b + c + d = 4, does d = 1? [#permalink]

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New post 21 Oct 2017, 01:08
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arabella wrote:
If for integers a, b, c, and d, a + b + c + d = 4, does d = 1?

(1) abc = 1
(2) ad = 1


\(d= 4-(a+b+c)\)

Statement 1: as all are integers so this implies that \(|a|=|b|=|c|=1\)

now if all are positive then \(d=4-(1+1+1)=1\)

but if any two are negative and the 3rd is positive then \(d=4-(1-1-1)=5\). Hence insufficient

Statement 2: this implies that \(|a|=|d|=1\). but \(d\) can be positive or negative. hence no unique value. Insufficient

Combining 1 & 2, if \(a=1\), then \(d=1\)

but if \(a=-1\), then either \(b\) or \(c\) is \(-1\) and the remaining is \(1\) for Statement 1 to hold true

so \(d=4-(-1-1+1)=5\)

but from statement 2, if \(a=-1\) then \(d=-1\). so this implies that \(a\) is not equal to \(-1\). Hence \(a=d=1\) Sufficient

Option C

Last edited by niks18 on 21 Oct 2017, 01:26, edited 3 times in total.

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Re: If for integers a, b, c, and d, a + b + c + d = 4, does d = 1? [#permalink]

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New post 21 Oct 2017, 01:12
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statement 1

abc =1
=>either of 2 variables are negative and = -1,

Then for a+b+c+d = 4

-2+1+d = 4
=> d = 5

or all of them are 1 (As they can't be fractions -> mentioned Integers)
=> 3+d= 4
=>d = 1

We have 2 sol. d can be 1 or 5
_____________________________________________________

statement 2

ad= 1

=> a, d = 1 or a, d = -1

We have 2 sol, d can be 1 or -1
_________________________________

Combining both eq.
we have d =1

Answer is C: Both statement sufficient.

Kudos [?]: 6 [1], given: 93

Re: If for integers a, b, c, and d, a + b + c + d = 4, does d = 1?   [#permalink] 21 Oct 2017, 01:12
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