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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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Updated on: 21 Oct 2017, 00:02
00:00

Difficulty:

75% (hard)

Question Stats:

43% (01:13) correct 57% (02:00) wrong based on 41 sessions

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

(1) abc = 1

Originally posted by arabella on 20 Oct 2017, 23:53.
Last edited by Bunuel on 21 Oct 2017, 00:02, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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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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Updated on: 21 Oct 2017, 00:26
1
arabella wrote:
If for integers a, b, c, and d, a + b + c + d = 4, does d = 1?

(1) abc = 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

Originally posted by niks18 on 21 Oct 2017, 00:08.
Last edited by niks18 on 21 Oct 2017, 00:26, 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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21 Oct 2017, 00:12
1
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

=> 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.
Re: If for integers a, b, c, and d, a + b + c + d = 4, does d = 1? &nbs [#permalink] 21 Oct 2017, 00:12
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