Integer problem 1

What day of the week is today? Then, what day will it be, 2³⁰ days from today?

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Solution 1

Calculating 2³⁰

  2³⁰
=(1000+24)³
=1000³+3*24*1000²+3*24²*1000+24³
=1000³+72*1000²+1728*1000+13824
=1073741824
=7*153391689+1.
Hence, the remainder of 2³⁰ divided by 7 is 1.
For example, if today is Sunday, 2³⁰ days from today will be Monday.

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Solution 2

Examining the remainders of 2, 2², 2³, ... divided by 7s

The remainders of 2, 4, 8, 16, 32, 64, ... divided by 7s are 2, 4, 1, 2, 4, 1, ... respectively.
Suppose the remainder of 2^k is 'r', or 2^k=7n+r, (Note: 'k', 'n', and 'r' are nonnegative integers.)
  2^k+3
=2³(7n+r)
=7(8n+r)+r.
Therefore, the remainder of 2^k+3 divided by 7 is equal to the remainder of 2^k divided by 7.
This means that the remainder of 2³⁰ divided by 7 is, eventually, the remainder of 2³ divided by 7, which is 1.
Hence, if today is Tuesday, for example, 2³⁰ days from today will be Wednesday.

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Solution 3

Utilizing 2³⁰=8¹⁰

  2³⁰
=8¹⁰
=(7+1)¹⁰
=7¹⁰+10*7⁹+...+10*7+1
=7n+1. (Note: 'n' is an integer.)
Hence, the remainder of 2³⁰ divided by 7 is 1.
For example, if today is Thursday, 2³⁰ days from today will be Friday.

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Solution 4

Factorizing 2³⁰-1

  2³⁰
=8¹⁰
=(8¹⁰-1)+1
=(8-1)(8⁹+8⁸+...+8+1)+1
=7n+1. (Note: 'n' is an integer.)
Hence, the remainder of 2³⁰ divided by 7 is 1.
For example, if today is Saturday, 2³⁰ days from today will be Sunday.

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Mathematical Problems

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© 2006 Takashi Shimazaki
Updated: April 14, 2013