The question asks us to find the base of a number system given an arithmetic equation: 52/4 = 12.

Understanding Number System Bases

In any number system with base 'b', a number represented as $d_n d_{n-1} ... d_1 d_0$ has a value in base 10 calculated as: $d_n \times b^n + d_{n-1} \times b^{n-1} + ... + d_1 \times b^1 + d_0 \times b^0$. The digits used in base 'b' must be less than 'b'. In our problem, the digits used are 1, 2, 4, and 5. Therefore, the base 'b' must be greater than 5.

Converting the Equation to Base 10

Let's assume the base is 'b'. We need to convert each number in the equation 52/4 = 12 from base 'b' to base 10:

• The number 52 in base 'b' ($52_b$) is equal to $5 \times b^1 + 2 \times b^0$ in base 10. This simplifies to $5b + 2$.
• The number 4 in base 'b' ($4_b$) is equal to $4 \times b^0$ in base 10. This is simply 4 (since b > 4).
• The number 12 in base 'b' ($12_b$) is equal to $1 \times b^1 + 2 \times b^0$ in base 10. This simplifies to $b + 2$.

Solving the Equation for Base 'b'

Now we can rewrite the given equation 52/4 = 12 in base 10 using the converted values:

To solve for 'b', we can perform the following algebraic steps:

1. Multiply both sides by 4: $$ 5b + 2 = 4(b + 2) $$
2. Distribute the 4 on the right side: $$ 5b + 2 = 4b + 8 $$
3. Subtract $4b$ from both sides: $$ 5b - 4b + 2 = 8 $$ $$ b + 2 = 8 $$
4. Subtract 2 from both sides: $$ b = 8 - 2 $$ $$ b = 6 $$

Verification

We found the base $b=6$. Let's check if this base is valid and if the original equation holds true.

• Validity: The digits used (1, 2, 4, 5) are all less than 6. So, the base $b=6$ is valid.
• Equation Check:
  • $52_6 = (5 \times 6^1) + (2 \times 6^0) = 30 + 2 = 32_{10}$
  • $4_6 = 4_{10}$
  • $12_6 = (1 \times 6^1) + (2 \times 6^0) = 6 + 2 = 8_{10}$
  Is $32_{10} / 4_{10} = 8_{10}$? Yes, $32 / 4 = 8$. The equation holds true.

Therefore, the base of the number system is 6.