Question

question 7 of 10
one column of numbers consists of 61, 24, and 47. when the digits of the
numbers are added together, the result is 6 + 1 + 2 + 4 + 4 + 7 = 24, and when
the digits of 24 are then added together, the end result is 2 + 4 = 6. if the
same process is performed on the numbers in a second column, what can be
concluded?

a. if the end result from the second column is also 6, then the sum of
the numbers in the first column is not equal to the sum of the
numbers in the second column.

b. if the end result from the second column is not 6, then the sum of
the numbers in the first column is equal to the sum of the
numbers in the second column.

c. if the end result from the second column is not 6, then the sum of
the numbers in the first column is not equal to the sum of the
numbers in the second column.

d. if the end result from the second column is also 6, then the sum of
the numbers in the first column is equal to the sum of the
numbers in the second column.

Explanation:

1. First, understand the process: The process described is finding the digital root (repeatedly summing digits until a single - digit number is obtained). For the first column, the sum of the numbers is \(61 + 24+47=132\), and the digital root of 132: \(1 + 3+2 = 6\). The digital root of a number is congruent to the number modulo 9. So, if two sets of numbers have the same digital root, their sums are congruent modulo 9, but not necessarily equal. However, if their digital roots are different, their sums are not congruent modulo 9, and thus cannot be equal.
2. Analyze Option A: If the second column also has a digital root of 6, the sums could be equal (e.g., if the second column has numbers that also sum to 132) or different (e.g., the second column has numbers that sum to \(132 + 9=141\), whose digital root is also 6). So, we cannot conclude that the sums are not equal.
3. Analyze Option B: If the second column's digital root is not 6, their sums are not congruent modulo 9, so they cannot be equal. So, this option is wrong.
4. Analyze Option C: As the digital root of a number is equal to the number modulo 9, if the digital roots (end results) are different, the numbers (sums of the columns) are not congruent modulo 9, and thus the sums cannot be equal. This option is correct.
5. Analyze Option D: Just because the digital roots are the same (both 6), the sums can be different (as shown in the analysis of Option A). So, we cannot conclude that the sums are equal.

Answer:

C. If the end result from the second column is not 6, then the sum of the numbers in the first column is not equal to the sum of the numbers in the second column.