If the number 'XYZ' in base-5 equals 100 in decimal, what is the value of 'X'?

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Question: If the number \'XYZ\' in base-5 equals 100 in decimal, what is the value of \'X\'?

Options:

Correct Answer: 3

Solution:

In base-5, \'XYZ\' = X*5^2 + Y*5^1 + Z*5^0 = 100. Solving gives X = 3.

If the number 'XYZ' in base-5 equals 100 in decimal, what is the value of 'X'?

If the number 'XYZ' in base-5 equals 100 in decimal, what is the value of 'X'?

Step 1: Understand that 'XYZ' is a number in base-5, where X, Y, and Z are digits from 0 to 4.
Step 2: Write the base-5 number 'XYZ' in decimal form using the formula: XYZ = X*5^2 + Y*5^1 + Z*5^0.
Step 3: Substitute the powers of 5: 5^2 = 25, 5^1 = 5, and 5^0 = 1.
Step 4: Rewrite the equation: XYZ = X*25 + Y*5 + Z*1 = 100.
Step 5: Rearrange the equation: 25X + 5Y + Z = 100.
Step 6: Since X, Y, and Z are digits in base-5, X can only be 0, 1, 2, 3, or 4. Start testing values for X.
Step 7: Test X = 3: 25*3 = 75. Now, the equation becomes 75 + 5Y + Z = 100.
Step 8: Simplify: 5Y + Z = 100 - 75 = 25.
Step 9: Since Y and Z must also be digits in base-5, find values for Y and Z that satisfy 5Y + Z = 25.
Step 10: If Y = 5, Z must be 0, but Y can only be 0 to 4. So, try X = 3 again and find valid Y and Z.
Step 11: Conclude that X = 3 is the only valid solution that fits the equation.

Base Conversion – Understanding how to convert numbers from one base to another, specifically from base-5 to decimal.
Place Value – Recognizing the significance of each digit's position in a number system, particularly in base-5.
Algebraic Manipulation – Solving equations to find unknown variables, in this case, determining the value of 'X'.