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# If a, b, c and d are four consecutive integers, and a^b = c^d

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If a, b, c and d are four consecutive integers, and a^b = c^d  [#permalink]

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Updated on: 10 Oct 2019, 15:51
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If $$a$$, $$b$$, $$c$$ and $$d$$ are four consecutive integers (not necessarily in that order), and $$a^b = c^d$$, what is the least possible value of $$a+b+c+d$$ ?

A) -8
B) -6
C) -2
D) 6
E) 12

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Originally posted by GMATPrepNow on 01 Oct 2019, 08:28.
Last edited by GMATPrepNow on 10 Oct 2019, 15:51, edited 1 time in total.
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Re: If a, b, c and d are four consecutive integers, and a^b = c^d  [#permalink]

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01 Oct 2019, 08:34
2^0=1^3
So, 0+1+2+3=6
Option D

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Re: If a, b, c and d are four consecutive integers, and a^b = c^d  [#permalink]

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Updated on: 01 Oct 2019, 15:12
integers can be x, x+1 , x+2 & x+3

sum of integers ; 4x+6 ; 4(x+1)+2
answer can be multiple of 4 ; IMO A ; -8

GMATPrepNow wrote:
If $$a$$, $$b$$, $$c$$ and $$d$$ are four consecutive integers (not necessarily in that order), and $$a^b = c^d$$, what is the least possible value of $$a+b+c+d$$ ?

A) -8
B) -6
C) -2
D) 6
E) 12

Originally posted by Archit3110 on 01 Oct 2019, 08:51.
Last edited by Archit3110 on 01 Oct 2019, 15:12, edited 1 time in total.
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Re: If a, b, c and d are four consecutive integers, and a^b = c^d  [#permalink]

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01 Oct 2019, 10:11
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Hint: The correct answer is not D
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Re: If a, b, c and d are four consecutive integers, and a^b = c^d  [#permalink]

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01 Oct 2019, 10:16
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Is the correct answer B

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Re: If a, b, c and d are four consecutive integers, and a^b = c^d  [#permalink]

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01 Oct 2019, 10:37
a=-1 b=0 c=1 d=-2

a^b=-1^0=1

c^d= 1^-2= 1

thus -2+-1+0+1 = -2

Ans-C

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Re: If a, b, c and d are four consecutive integers, and a^b = c^d  [#permalink]

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01 Oct 2019, 11:08
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GMATPrepNow wrote:
If $$a$$, $$b$$, $$c$$ and $$d$$ are four consecutive integers (not necessarily in that order), and $$a^b = c^d$$, what is the least possible value of $$a+b+c+d$$ ?

A) -8
B) -6
C) -2
D) 6
E) 12

This question takes some playing around with numbers.
It also helps to recognize the following properties:

$$1^k = 1$$ for all values of k

$$(-1)^k = 1$$ for all EVEN integer values of k

$$k^0 = 1$$ for all non-zero values of k

If $$a = -1$$, $$b = -2$$, $$c=-3$$ and $$d = 0$$, then the equation $$a^b = c^d$$ becomes $$(-1)^{-2} = (-3)^0$$

Simplify to get: $$1= 1$$....PERFECT!

In this case, $$a+b+c+d=(-1)+(-2)+(-3)+0=-6$$

So, it's possible to get a sum of -6 (answer choice B).
HOWEVER, perhaps it's possible to get a sum of -8 (answer choice A)
Well, it turns out that we CAN'T get a sum of -8. Here's why:

We know that the 4 numbers are CONSECUTIVE INTEGERS.
So, if we let x = the smallest integer, then
x+1 = the next integer
x+2 = the next integer
And x+3 = the next integer

So, the sum of these 4 consecutive integers = x + (x+1) + (x+2) + (x+3)
= 4x + 6
= 4x + 4 + 2
= 4(x+1) + 2
Notice that 4(x+1) is a multiple of 4
So, 4(x+1) + 2 is 2 greater than some multiple of 4

Since the sum of the four integers must be 2 greater than some multiple of 4, we
can eliminate answer choice A, since -8 is a multiple of 4

Cheers,
Brent
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Re: If a, b, c and d are four consecutive integers, and a^b = c^d  [#permalink]

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10 Oct 2019, 10:50
Can someone please post the correct method of solving this question.
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Re: If a, b, c and d are four consecutive integers, and a^b = c^d  [#permalink]

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10 Oct 2019, 15:39
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sp4182 wrote:
Can someone please post the correct method of solving this question.

I know the above solutions aren't exactly algorithmic (like many/most GMAT quant questions), but there are still plenty of official questions that require test-takers to find values that meet a certain set of conditions.
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Re: If a, b, c and d are four consecutive integers, and a^b = c^d  [#permalink]

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10 Oct 2019, 15:49
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Dear Brent GMATPrepNow,

If answer is B, why do you mark E under the spoiler? I think there is mistake.
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Re: If a, b, c and d are four consecutive integers, and a^b = c^d  [#permalink]

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10 Oct 2019, 15:52
Top Contributor
Mo2men wrote:
Dear Brent GMATPrepNow,

If answer is B, why do you mark E under the spoiler? I think there is mistake.

Thanks for pointing that out!!
The correct answer is, indeed, B.
I changed the OA.

Kudos for you!!!

Cheers,
Brent
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Re: If a, b, c and d are four consecutive integers, and a^b = c^d   [#permalink] 10 Oct 2019, 15:52
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# If a, b, c and d are four consecutive integers, and a^b = c^d

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