The problem asks to identify which of the following number conversions is true: A. $11001_2 = 39_{10}$ B. $31_8 = 11001_2$ C. $63_{10} = 122_5$ D. $24_5 = 13_8$

1. Problem Description

The problem asks to identify which of the following number conversions is true:

A. $11001_2 = 39_{10}$

B. $31_8 = 11001_2$

C. $63_{10} = 122_5$

D. $24_5 = 13_8$

2. Solution Steps

A. $11001_2 = 1 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 = 16 + 8 + 0 + 0 + 1 = 25_{10}$. Since $25 \neq 39$, option A is incorrect.

B. $31_8 = 3 \cdot 8^1 + 1 \cdot 8^0 = 24 + 1 = 25_{10}$. From A, $11001_2 = 25_{10}$. Thus, $31_8 = 11001_2 = 25_{10}$. Option B is correct.

C. $63_{10}$. $122_5 = 1 \cdot 5^2 + 2 \cdot 5^1 + 2 \cdot 5^0 = 25 + 10 + 2 = 37_{10}$. Since $63 \neq 37$, option C is incorrect.

D. $24_5 = 2 \cdot 5^1 + 4 \cdot 5^0 = 10 + 4 = 14_{10}$. $13_8 = 1 \cdot 8^1 + 3 \cdot 8^0 = 8 + 3 = 11_{10}$. Since $14 \neq 11$, option D is incorrect.

3. Final Answer

B. $31_8 = 11001_2$