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For which of the following values of x is the result an integer when

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mdmarufhasan
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For which of the following values of x is the result an integer when [#permalink] New post   Updated on: 04 Jan 2023, 02:29
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For which of the following values of x is the result an integer when (2^20 - x) is divided by 3?

I. 0
II. 1
III. 4

A. I only
B. II only
C. III only
D. I and II only
E. II and III only
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E

Originally posted by mdmarufhasan on 03 Jan 2023, 20:24.
Last edited by Bunuel on 04 Jan 2023, 02:29, edited 1 time in total.
Renamed the topic and edited the question.
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Re: For which of the following values of x is the result an integer when [#permalink] New post   04 Jan 2023, 02:36
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mdmarufhasan wrote:
For which of the following values of x is the result an integer when (2^20 - x) is divided by 3?

I. 0
II. 1
III. 4

A. I only
B. II only
C. III only
D. I and II only
E. II and III only


Let's check:

I. 0

    2^20 - 0 = 2^20, which clearly is not a multiple of 3 (2^20 has only 2's in its prime factorization).


II. 1

    2^20 - 1 = (2^10 - 1)(2^10 + 1) = 1023*1025. Now, this product is divisible by 3 because 1023 is divisible by 3 (the sum of its digits is a multiple of 3), then

III. 4

    2^20 - 4 = (2^10 - 2)(2^10 + 2) = 1022*1026. Now, this product is divisible by 3 because 1026 is divisible by 3 (the sum of its digits is a multiple of 3), then

Answer: E (II and III only).

Hope it helps.
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For which of the following values of x is the result an integer when [#permalink] New post   05 Jan 2023, 20:41
Asked: For which of the following values of x is the result an integer when (2^20 - x) is divided by 3?

2^20 = 32^4 = (33-1)^4
The remainder when 2^20 = (33-1)^4 is divided by 3 = (-1)^4 = 1

Quote:
I. 0

The remainder when 2^20 - 0 = (33-1)^4 - 0 is divided by 3 = 1-0 = 1:
The result an integer when 2^20 - 1 = (33-1)^4 - 1 is divided by 3 = k + 1/3: where k is an integer

Quote:
II. 1

The remainder when 2^20 - 1 = (33-1)^4 - 1 is divided by 3 = (1 - 1)/3 = 0
The result an integer when 2^20 - 1 = (33-1)^4 - 1 is divided by 3 = k : where k is an integer

Quote:
III. 4

The remainder when 2^20 - 4 = (33-1)^4 - 4 is divided by 3 =(1 + 3 - 4)/3 = 0: INTEGER
The result an integer when 2^20 - 1 = (33-1)^4 - 1 is divided by 3 = k : where k is an integer

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

IMO E
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Re: For which of the following values of x is the result an integer when [#permalink] New post   07 Jan 2023, 17:50
mdmarufhasan wrote:
For which of the following values of x is the result an integer when (2^20 - x) is divided by 3?

I. 0
II. 1
III. 4

A. I only
B. II only
C. III only
D. I and II only
E. II and III only




2^20 = 4^10 => (3+1)^10 => It will be in the form of 3k + 1

Hence x has to be in 3y+1 form to make the expression divisible by 3.

IMO E
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Re: For which of the following values of x is the result an integer when [#permalink]
07 Jan 2023, 17:50
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Bunuel
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