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If x is a positive integer, is x^5 + 1 an odd number?

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Bunuel
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If x is a positive integer, is x^5 + 1 an odd number? [#permalink] New post   06 Jun 2023, 01:30
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A
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C
D
E

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If x is a positive integer, is x^5 + 1 an odd number?

(1) x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.
(2) 5^x is an odd number


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Re: If x is a positive integer, is x^5 + 1 an odd number? [#permalink] New post   06 Jun 2023, 01:57
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S1) x is the LCM of (21,22,23,24)
Hence x = 2^3 * 3 * 7 * 11 * 23
The presence of 2 indicates that x is even.

Hence x^5 + 1 = odd
So S1) is sufficient.
Eliminate options C & E.

S2) 5^x = odd.
Note: 5 is a base which will always yield odd numbers as long as it has positive integers as powers.
so 5^1 = odd where x = odd, hence x^5 + 1 = even
but 5^2=odd, where x = even, hence x^5+1 = odd.

Thus S2, is insufficient. Eliminate options B and D.

Answer is A.
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Re: If x is a positive integer, is x^5 + 1 an odd number? [#permalink] New post   06 Jun 2023, 02:02
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If x is a positive integer, is x^5 + 1 an odd number?

(1) x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.

21,22,23, and 24 are factors of x. This essentially means that x is LCM of these 4.
X= 2^3*3*7*11*23
X is an even number.
X^5 too will be an even number.
X^5+1 will be an odd number.

We are getting a unique answer.

Statement I alone is sufficient.
Reject B,C, and E.

(2) 5^x is an odd number

X=1, 5^1=5 (odd)
X^5+1= 1+1=2 (even).....NO
X=2, 5^2= 25 (odd)
X^5+1= 32+1=33 (odd).....YES

We aren't getting a unique answer this time.
Reject D.

Option A is the correct answer

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Re: If x is a positive integer, is x^5 + 1 an odd number? [#permalink] New post   06 Jun 2023, 02:36
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for x^5+1 to be odd x has to be even
#1
x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.
x will be 2^3*3*7*11*23 ; even
sufficient
#2
5^x is an odd number
x can be any integer value even or odd
insufficient
option A


Bunuel wrote:
If x is a positive integer, is x^5 + 1 an odd number?

(1) x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.
(2) 5^x is an odd number


This is a Data Sufficiency Butler Question

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gmatclubot
Re: If x is a positive integer, is x^5 + 1 an odd number? [#permalink]
06 Jun 2023, 02:36
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Bunuel
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