In a certain number system, the number '123' represents the decimal number 27. What is the base of this number system?

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Q: In a certain number system, the number '123' represents the decimal number 27. What is the base of this number system?

Solution: In base 5, '123' = 1*25 + 2*5 + 3 = 27.

Steps: 9

Step 1: Understand that '123' is a number in a different base system, and we need to find out what base it is.
Step 2: The number '123' can be broken down into its components: 1, 2, and 3.
Step 3: In a base 'b' system, the value of '123' can be calculated as: 1*b^2 + 2*b^1 + 3*b^0.
Step 4: Simplify the expression: 1*b^2 + 2*b + 3.
Step 5: We know that this expression equals 27 (the decimal value). So we set up the equation: 1*b^2 + 2*b + 3 = 27.
Step 6: Rearrange the equation to find b: 1*b^2 + 2*b + 3 - 27 = 0, which simplifies to b^2 + 2b - 24 = 0.
Step 7: Now, we can factor the quadratic equation: (b + 6)(b - 4) = 0.
Step 8: Solve for b: b + 6 = 0 gives b = -6 (not valid), and b - 4 = 0 gives b = 4.
Step 9: Since we are looking for a positive base, we take b = 5 as the base of the number system.