Binary Sum

Number Theory Level 1

111111111111111 1 ( 2 ) + 101010101010101 0 ( 2 ) = ? 1111111111111111_{(2)} + 1010101010101010_{(2)} = \ ?

Clarification : The two numbers given are written in base 2, while the numbers in the answer choices are written in base 16.

1 B B B 9 ( 16 ) 1BBB9_{(16)} 1 C A A 9 ( 16 ) 1CAA9_{(16)} 1 D B B 9 ( 16 ) 1DBB9_{(16)} 1 A A A 9 ( 16 ) 1AAA9_{(16)}

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Farhabi Mojib by Farhabi Mojib , 23, Bangladesh

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2 solutions

Chew-Seong Cheong
Aug 25, 2015

1111 1111 1111 1111 ( 2 ) + 1010 1010 1010 1010 ( 2 ) = F F F F ( 16 ) + A A A A ( 16 ) = 1000 0 ( 16 ) − 1 ( 16 ) + A A A A ( 16 ) = 1 A A A A ( 16 ) − 1 ( 16 ) = 1 A A A 9 ( 16 ) \color{#D61F06} {1111} \space \color{#EC7300} {1111} \space \color{#20A900} {1111} \space \color{#3D99F6} {1111}_{(2)} + \color{#D61F06} {1010} \space \color{#EC7300} {1010} \space \color{#20A900} {1010} \space \color{#3D99F6} {1010}_{(2)} \\ = \color{#D61F06} {F} \space \color{#EC7300} {F} \space \color{#20A900} {F} \space \color{#3D99F6} {F}_{(16)} + \color{#D61F06} {A} \space \color{#EC7300} {A} \space \color{#20A900} {A} \space \color{#3D99F6} {A}_{(16)} \\ = 10000_{(16)} - 1_{(16)} + AAAA_{(16)} \\ = 1AAAA_{(16)} - 1_{(16)}\\ = \boxed{1AAA9_{(16)}}

47 Helpful 3 Interesting 6 Brilliant 0 Confused

Moderator note:

Yes. That's how you should solve problems like these. The colors is a nice bonus as well. Thanks!

Wow, rainbow. Liked your organized solution though. Thanks

Joshua Yoo - 5 years, 9 months ago
Hadia Qadir
Aug 30, 2015

f we look at it carefully, the first number is 16 "1"s while the second is 8 repeating "10"s. We can break the big number down into their respective subunits and consider: 11 + 10 in binary gives 101.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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