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

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

Step 1: Understand that 'XYZ' in base 5 means each letter represents a digit in base 5.
Step 2: Recognize that in base 5, the place values are powers of 5. The leftmost digit (X) is multiplied by 5^2 (which is 25), the middle digit (Y) is multiplied by 5^1 (which is 5), and the rightmost digit (Z) is multiplied by 5^0 (which is 1).
Step 3: Write the equation based on the place values: X*25 + Y*5 + Z = 100.
Step 4: Since X, Y, and Z must be digits in base 5, they can only be 0, 1, 2, 3, or 4.
Step 5: Start by finding the maximum value for X. If X = 4, then 4*25 = 100, which means Y and Z must be 0. This is valid but we need to check lower values of X.
Step 6: If X = 3, then 3*25 = 75. This leaves 100 - 75 = 25. The maximum value for Y*5 + Z = 25 is not possible since Y can only be 4 (4*5 = 20) and Z can only be 4 (4*1 = 4), totaling 24.
Step 7: If X = 2, then 2*25 = 50. This leaves 100 - 50 = 50. The maximum value for Y*5 + Z = 50 is not possible since Y can only be 4 (4*5 = 20) and Z can only be 4 (4*1 = 4), totaling 24.
Step 8: If X = 1, then 1*25 = 25. This leaves 100 - 25 = 75. The maximum value for Y*5 + Z = 75 is not possible since Y can only be 4 (4*5 = 20) and Z can only be 4 (4*1 = 4), totaling 24.
Step 9: If X = 0, then 0*25 = 0. This leaves 100 - 0 = 100. This is not possible since Y and Z cannot be greater than 4.
Step 10: The only valid solution is when X = 2.

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 Representation – Using algebra to represent the relationship between the digits and their corresponding values in decimal.