Finding the Minimum Value of a * b for Given Base Conversions

Consider the problem where the representation of a number in base a is 123a, and in base b is 146b. We aim to determine the minimum value of a * b that satisfies this condition. This exploration involves several steps, including converting the numbers from non-decimal bases to base 10, setting up and solving a corresponding equation, and testing integer values for a and b.

Conversion to Base 10

To start, we convert the given numbers from their respective bases to the decimal system, which is the base of our arithmetic operations.

Step 1: Conversion of 123a to Base 10

The number 123a can be expressed in base 10 as:

[123_a 1 cdot a^2 2 cdot a 3]

Step 2: Conversion of 146b to Base 10

The number 146b can be expressed in base 10 as:

[146_b 1 cdot b^2 4 cdot b 6]

Setting Up the Equation

We then set these two expressions equal to each other, as they represent the same number in different bases:

[1 cdot a^2 2 cdot a 3 1 cdot b^2 4 cdot b 6]

By simplifying and rearranging, we obtain a quadratic equation:

[a^2 - b^2 2a - 4b - 3 0]

This equation can be factored as:

[(a - b)(a b) - 2(a - b) - 3 (a - b)(a b - 2) - 3 0]

Testing Integer Values for a and b

Since a and b must be greater than the digits they represent, we start with the smallest possible values, a > 3 and b > 6. We will test several integer values of a and b to find the smallest pair that satisfies the equation.

Example Calculations

For (a 4, b 7)

[123_4 1 cdot 4^2 2 cdot 4 3 16 8 3 27]

[146_7 1 cdot 7^2 4 cdot 7 6 49 28 6 83]

We find these values do not match, so we continue testing.

For (a 5, b 8)

[123_5 1 cdot 5^2 2 cdot 5 3 25 10 3 38]

[146_8 1 cdot 8^2 4 cdot 8 6 64 32 6 102]

Again, these values do not match.

For (a 6, b 9)

[123_6 1 cdot 6^2 2 cdot 6 3 36 12 3 51]

[146_9 1 cdot 9^2 4 cdot 9 6 81 36 6 123]

These values match, but we have to ensure this is the smallest valid pair. We continue testing.

For (a 7, b 8)

[123_7 1 cdot 7^2 2 cdot 7 3 49 14 3 66]