Which of the following is NOT a valid octal number?

The correct answer is D, E. Hexadecimal has base 16, and because 2 raised to the power 4 equals 16, exactly four binary bits make one hexadecimal digit. The given number 1110 is already a group of four bits, so it converts in one step. Its place values are 8, 4, 2 and 1, and the bits are 1, 1, 1 and 0, so the decimal value is 8 + 4 + 2 + 0, that is 14. In hexadecimal the decimal values 10 to 15 are written as the letters A to F, so 10 is A, 11 is B, 12 is C, 13 is D, 14 is E and 15 is F. Fourteen is therefore E. Option A is wrong because F stands for 15, whose binary form is 1111. Option B is wrong because B stands for 11, that is 1011. Option C is wrong because A stands for 10, that is 1010.

The correct answer is C, 110010111. Octal has base 8, and because 2 raised to the power 3 equals 8, every octal digit expands into exactly three binary bits. Take the digits of 627 one at a time. The digit 6 becomes 110, the digit 2 becomes 010 and the digit 7 becomes 111. Writing them in the same order gives 110 010 111, that is 110010111. A quick check works the other way: grouping 110010111 in threes from the right gives 110, 010 and 111, which read back as 6, 2 and 7. Option A is wrong because 111010110 reads as 7, 2 and 6, which is octal 726. Option B is wrong because 110010110 ends in 110 and so reads as 626. Option D is wrong because 011010111 reads as 3, 2 and 7, that is octal 327.

The correct answer is A, (321)8. To turn a binary number into octal, group the bits in threes starting from the rightmost bit, adding leading zeros to complete the last group, because three bits cover the eight octal digits. The number 11010001 splits into 11, 010 and 001, and the first group is written as 011. The group 011 has the value 3, the group 010 has the value 2 and the group 001 has the value 1, so the octal number is 321. Checking in decimal confirms it: 11010001 is 128 + 64 + 16 + 1, which equals 209, and octal 321 is 3 times 64 plus 2 times 8 plus 1, again 209. Option B is wrong because 123 is the same digits reversed. Option C is wrong because 641 would need the bits 110100001. Option D is wrong because 146 in octal equals 102 in decimal.

The correct answer is C, 10111. The safest way to handle a binary subtraction in an examination is to convert both numbers to decimal, subtract, and convert the answer back. The first number 10010000 has ones in the places worth 128 and 16, so it equals 144. The second number 1111001 has ones in the places worth 64, 32, 16, 8 and 1, so it equals 121. The difference is 144 minus 121, which is 23. Now convert 23 to binary by repeated division by 2, which gives the remainders 1, 1, 1, 0 and 1 read upwards, that is 10111. Option A is wrong because 11101 equals 29. Option B is wrong because 11011 equals 27. Option D is wrong because 10011 equals 19. Only 10111 equals 23, so C is the answer.

The correct answer is B, 4. A bit, short for binary digit, is a single 0 or 1 and is the smallest unit of data a computer handles. Four bits together are called a nibble, which is exactly half of a byte, and the playful name comes from a nibble being a small bite. The nibble matters in practice because one hexadecimal digit is stored in exactly four bits, which is why hexadecimal is such a convenient shorthand for binary. Option A is wrong because two bits form no named unit and can represent only four combinations. Option C is wrong because eight bits make one byte, the unit in which a single character is normally stored. Option D is wrong because sixteen bits make two bytes, often called a word on older machines. Remember the ladder: bit, nibble of 4 bits, byte of 8 bits.