To solve the problem of converting the decimal number (9_{10}) to its equivalent in octal (base 8), we need to follow a systematic approach. Let's break it down step-by-step.
Step 1: Understanding the Number Systems
- Decimal (Base 10): This is the standard number system that we use every day. It consists of digits from 0 to 9.
- Octal (Base 8): This number system uses digits from 0 to 7.
Step 2: Converting Decimal to Octal
To convert a decimal number to octal, we can use repeated division by 8 and keep track of the remainders. Here’s how it works for (9_{10}):
- Divide the decimal number by 8:
- (9 \div 8 = 1) with a remainder of (1).
Here, we take the integer part of the division (which is (1)) and the remainder (which is (1)).
- Continue dividing the quotient by 8:
- Now take the quotient (1) and divide it by (8):
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(1 \div 8 = 0) with a remainder of (1).
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Stop when the quotient is 0:
- We stop here because the quotient is now (0).
Step 3: Collecting the Remainders
Now, we collect the remainders from the last division to the first:
- The first remainder (from the first division) is (1).
- The second remainder (from the second division) is also (1).
So, when we read the remainders from bottom to top, we get (11_8).
Conclusion
Thus, (9_{10}) is equivalent to (11_8). Therefore, (X) in the expression (X_8) is (11).
Answer
The correct option is
C. 11.
Explanation of Other Options
- A. 7: This is incorrect because (7_{10}) is less than (9_{10}) and does not represent the same value in octal.
- B. 10: This is also incorrect. (10_8) is equal to (8_{10}), which is not equivalent to (9_{10}).
- D. 12: This is incorrect as well. (12_8) is equal to (10_{10}), which is greater than (9_{10}).
Revision Summary
- To convert from decimal to octal, divide the decimal number by 8 and record the remainders.
- Continue dividing until the quotient is 0, then read the remainders in reverse order.
- (9_{10}) converts to (11_8), making the answer (X = 11).
- Always check the options against the conversion to ensure accuracy.