The decimal number system uses the characters 0, 1, 2, …, 8, 9, and the octal number system uses the characters 0, 1, 2, …, 6, 7.
For example, the decimal number 12 ( ) is expressed as 14 ( ) in the octal number system.
The decimal number 108 in the octal number system is
168
108
150
154
ANSWER
(D) 154. Dividing 108 by 8 over and over leaves remainders 4, 5, 1, so . Check: .
Official answer: DSource: Official answer key, GATE 2024 EE (IISc Bengaluru)
To convert a decimal number to octal, divide by 8 repeatedly and note each remainder. Read from last to first, the remainders are the octal digits.
with remainder (since ), so the units digit is .
with remainder , so the eights digit is .
with remainder , so the sixty-fours digit is .
Reading the remainders from last to first gives .
Check using the place values (powers of 8):
Ruling out the other options: an octal numeral can only use the digits 0 to 7, so (A) 168 is not a valid octal number. (B) is just the decimal numeral copied unchanged; read as octal it means . (C) . Only (D) gives 108.
KEY IDEA
In octal the place values are powers of 8 (, , ). To convert, divide by 8 repeatedly and read the remainders from bottom to top. Then expand the result in powers of 8 to confirm it gives back the decimal number.
WATCH OUT FOR
Reading the remainders top to bottom gives 451 instead of 154. Another slip is choosing 168 or 108 without noticing that 168 uses the digit 8, which octal does not have, and that 108 is just the decimal numeral copied unchanged.
From the GATE 2024 EE paper. Every question of this paper · Download the full paper · All 58 Numerical questions, with notes
