If 'XYZ' in base-3 equals 26 in decimal, what is the value of 'X'?

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Question: If \'XYZ\' in base-3 equals 26 in decimal, what is the value of \'X\'?

Options:

Correct Answer: 2

Solution:

In base-3, \'XYZ\' means X*9 + Y*3 + Z = 26. The only valid digit for X is 2.

If 'XYZ' in base-3 equals 26 in decimal, what is the value of 'X'?

If 'XYZ' in base-3 equals 26 in decimal, what is the value of 'X'?

Step 1: Understand that 'XYZ' in base-3 means each letter represents a digit in base-3.
Step 2: In base-3, the digits can only be 0, 1, or 2.
Step 3: The position of each digit represents a power of 3: X is in the '3^2' position, Y is in the '3^1' position, and Z is in the '3^0' position.
Step 4: Write the equation for 'XYZ' in decimal: X*9 + Y*3 + Z = 26.
Step 5: Since X is the most significant digit, it must be the largest possible digit in base-3, which is 2.
Step 6: Substitute X = 2 into the equation: 2*9 + Y*3 + Z = 26.
Step 7: Calculate 2*9 = 18, so the equation becomes 18 + Y*3 + Z = 26.
Step 8: Simplify the equation: Y*3 + Z = 26 - 18, which gives Y*3 + Z = 8.
Step 9: Determine possible values for Y and Z that satisfy Y*3 + Z = 8, keeping in mind Y and Z can only be 0, 1, or 2.
Step 10: The only valid combination is Y = 2 and Z = 2, which means X = 2 is correct.

Base Conversion – Understanding how to convert numbers from one base to another, specifically from base-3 to decimal.
Place Value in Different Bases – Recognizing the significance of each digit's position in a number based on its base.
Digit Constraints in Base Systems – Knowing the valid digits for a given base, such as 0, 1, and 2 for base-3.