Convert 105(10) to binary.

Convert 105(10) to binary.

• A. 1101001(2)
• B. 1101000(2)
• C. 1101010(2)
• D. 1011011(2)

Answer: (A)

To convert a decimal number to binary, express it as a sum of powers of two and mark a 1 for each power that is used, with 0s for powers that aren’t. For 105, start with the largest power of two not exceeding it: 2^6 = 64. Subtracting leaves 41, so the bit for 2^6 is 1. Next, 2^5 = 32 fits into 41, leaving 9, so the bit for 2^5 is 1. Then 2^4 = 16 doesn’t fit into 9, so that bit is 0. 2^3 = 8 fits into 9, leaving 1, so the bit for 2^3 is 1. 2^2 = 4 doesn’t fit into 1, so that bit is 0. 2^1 = 2 doesn’t fit into 1, so that bit is 0. Finally, 2^0 = 1 fits, leaving 0, so the bit for 2^0 is 1. Reading the bits from 2^6 down to 2^0 gives 1101001, which is 105 in decimal. So the binary representation is 1101001(2). The other options do not equal 105 when evaluated in binary: 1101000(2) equals 104, 1101010(2) equals 106, and 1011011(2) equals 91.

To convert a decimal number to binary, express it as a sum of powers of two and mark a 1 for each power that is used, with 0s for powers that aren’t.

For 105, start with the largest power of two not exceeding it: 2^6 = 64. Subtracting leaves 41, so the bit for 2^6 is 1. Next, 2^5 = 32 fits into 41, leaving 9, so the bit for 2^5 is 1. Then 2^4 = 16 doesn’t fit into 9, so that bit is 0. 2^3 = 8 fits into 9, leaving 1, so the bit for 2^3 is 1. 2^2 = 4 doesn’t fit into 1, so that bit is 0. 2^1 = 2 doesn’t fit into 1, so that bit is 0. Finally, 2^0 = 1 fits, leaving 0, so the bit for 2^0 is 1.

Reading the bits from 2^6 down to 2^0 gives 1101001, which is 105 in decimal. So the binary representation is 1101001(2).

The other options do not equal 105 when evaluated in binary: 1101000(2) equals 104, 1101010(2) equals 106, and 1011011(2) equals 91.