If 'XYZ' in base-4 equals 27 in decimal, what is the value of 'X'?

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If 'XYZ' in base-4 equals 27 in decimal, what is the value of 'X'?

Step 1: Understand that 'XYZ' is a number in base-4, which means each digit represents a power of 4.
Step 2: Write down the formula for converting 'XYZ' from base-4 to decimal: XYZ = 4^2 * X + 4^1 * Y + 4^0 * Z.
Step 3: Recognize that 4^2 = 16, 4^1 = 4, and 4^0 = 1.
Step 4: Substitute these values into the formula: XYZ = 16 * X + 4 * Y + 1 * Z.
Step 5: Set the equation equal to 27, since 'XYZ' in decimal equals 27: 16 * X + 4 * Y + Z = 27.
Step 6: Since X, Y, and Z must be digits in base-4, they can only be 0, 1, 2, or 3.
Step 7: Test possible values for X, starting from the maximum (3) down to 0, to find valid combinations that satisfy the equation.
Step 8: When X = 3, substitute into the equation: 16 * 3 + 4 * Y + Z = 27, which simplifies to 48 + 4Y + Z = 27.
Step 9: This equation cannot hold true since 48 is already greater than 27, so try X = 2 next.
Step 10: When X = 2, substitute: 16 * 2 + 4 * Y + Z = 27, which simplifies to 32 + 4Y + Z = 27.
Step 11: This also cannot hold true since 32 is greater than 27, so try X = 1 next.
Step 12: When X = 1, substitute: 16 * 1 + 4 * Y + Z = 27, which simplifies to 16 + 4Y + Z = 27.
Step 13: Rearranging gives 4Y + Z = 11. Now test values for Y and Z that are valid in base-4.
Step 14: If Y = 2, then 4 * 2 = 8, and Z must be 3 to satisfy 8 + 3 = 11. This gives us X = 1, Y = 2, Z = 3.
Step 15: If Y = 3, then 4 * 3 = 12, which is too high since Z would need to be negative. So Y cannot be 3.
Step 16: The only valid combination found is X = 1, Y = 2, Z = 3, but we need to find the maximum value for X.
Step 17: Check if X = 3 is possible again, but it leads to invalid results. Therefore, the maximum valid X is 2.

Base Conversion – Understanding how to convert numbers from one base to another, specifically from base-4 to decimal.
Place Value – Recognizing the significance of each digit's position in a number system, particularly in base-4.
Digit Constraints – Knowing the valid digits in a base-4 system, which are 0, 1, 2, and 3.

If 'XYZ' in base-4 equals 27 in decimal, what is the value of 'X'?

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