1. The problem asks us to verify the equality $2078 = 31303_5$, where the subscript 5 indicates that $31303_5$ is a number in base 5. 2. First, convert the base 5 number $31303_5$ to base 10 (decimal). Each digit represents a power of 5, starting from the right with power 0: $$31303_5 = 3 \times 5^4 + 1 \times 5^3 + 3 \times 5^2 + 0 \times 5^1 + 3 \times 5^0$$ 3. Calculate each term: - $3 \times 5^4 = 3 \times 625 = 1875$ - $1 \times 5^3 = 1 \times 125 = 125$ - $3 \times 5^2 = 3 \times 25 = 75$ - $0 \times 5^1 = 0$ - $3 \times 5^0 = 3 \times 1 = 3$ 4. Sum all these values: $$1875 + 125 + 75 + 0 + 3 = 2078$$ 5. We see that $31303_5$ in decimal is $2078$, which matches the left side of the equation. 6. Therefore, the equality $2078 = 31303_5$ is true. Final answer: $2078 = 31303_5$ is correct.