1. The problem asks for the number that has the prime factor decomposition $5^2 \times 71$. 2. Prime factor decomposition means expressing a number as a product of prime numbers raised to their powers. 3. Here, the number is composed of the prime 5 raised to the power 2, and the prime 71 raised to the power 1. 4. To find the number, multiply these prime factors: $$5^2 \times 71 = (5 \times 5) \times 71 = 25 \times 71$$ 5. Calculate the product: $$25 \times 71 = 1775$$ 6. Therefore, the number with prime factor decomposition $5^2 \times 71$ is 1775.