(b) Does the calculation \( 8514+6872+2535+5961=23,782 \) have an error based on casting out nines? Select the correct choice below and fill in the answer box within your choice. A. The calculation checks using casting out nines. The remainder is \( \square \). B. Casting out nines shows there is an error. The given answer has a remainder of \( \square \), but casting out nines gives a remainder of \( \square \). The correct answer is \( \square \).

Let's break this down using casting out nines, a method that simplifies checking calculations by reducing numbers to their digit sums. For the individual numbers: - \( 8514 \) → \( 8 + 5 + 1 + 4 = 18 \) → \( 1 + 8 = 9 \) (remainder 0) - \( 6872 \) → \( 6 + 8 + 7 + 2 = 23 \) → \( 2 + 3 = 5 \) (remainder 5) - \( 2535 \) → \( 2 + 5 + 3 + 5 = 15 \) → \( 1 + 5 = 6 \) (remainder 6) - \( 5961 \) → \( 5 + 9 + 6 + 1 = 21 \) → \( 2 + 1 = 3 \) (remainder 3) Now summing these remainders: - Total remainder: \( 0 + 5 + 6 + 3 = 14 \) → \( 1 + 4 = 5 \) For the calculated answer \( 23,782 \): - \( 2 + 3 + 7 + 8 + 2 = 22 \) → \( 2 + 2 = 4 \) (remainder 4) Therefore, we find: - The remainder of the sum \( 8514+6872+2535+5961 \) is \( 5 \). - The remainder of the given answer \( 23,782 \) is \( 4 \). Since these remainders do not match, there is an error in the original calculation. So, your filled answer would be: B. Casting out nines shows there is an error. The given answer has a remainder of \( 4 \), but casting out nines gives a remainder of \( 5 \). The correct answer is \( 23,782 + 1 = 23,783 \).