(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 $$ \square $$.

Question

UpStudy AI · Accepted Answer

We first compute the digital root (or remainder when divided by 9) for each addend.

1. For $$8514$$:  
   $$
   8+5+1+4=18,\quad 18\mod 9=0.
   $$

2. For $$6872$$:  
   $$
   6+8+7+2=23,\quad 23\mod 9=23-18=5.
   $$

3. For $$2535$$:  
   $$
   2+5+3+5=15,\quad 15\mod 9=15-9=6.
   $$

4. For $$5961$$:  
   $$
   5+9+6+1=21,\quad 21\mod 9=21-18=3.
   $$

Adding these remainders we have:  
$$
0+5+6+3=14,\quad 14\mod 9=14-9=5.
$$

Thus, the sum should have a remainder of $$5$$ when divided by $$9$$.

Next, check the given answer $$23782$$:  
$$
2+3+7+8+2=22,\quad 22\mod9=22-18=4.
$$

Since the given answer has a remainder of $$4$$ (instead of $$5$$), casting out nines shows an error.

Additional verification by computing the actual sum:  
$$
8514+6872=15386,
$$
$$
15386+2535=17921,
$$
$$
17921+5961=23882.
$$

Thus, the correct answer should be $$23882$$.

Therefore, the answer is:

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 $$23882$$.
Casting out nines shows an error in the calculation. The given answer $$23,782$$ has a remainder of $$4$$, but it should have a remainder of $$5$$. The correct sum is $$23,882$$.

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