Question

DIRECTIONS for questions 45 and 46: Read the information below and answer the question that follows.
It is possible to arrange eight of the nine numbers 2, 3, 4, 5, 7, 10, 11, 12, 13 in the vacant squares of the 3 by 4 array shown below so that the arithmetic average of the numbers in each row and column is the same integer.

1			15
	9
		14

46. Which one of the nine numbers must be left out when completing the array?

• A. 4
• B. 5
• C. 7
• D. 10

Hint:

Find which single number, once removed from the sum of all thirteen entries in the grid, leaves a total divisible by 12.

Answer 1

The Correct Option is D

Step 1: Set up the total-sum condition.
The grid has 3 rows of 4 cells and 4 columns of 3 cells, all sharing one common average x. Summing the 3 rows (each summing to \(4x\)) covers the whole grid once, so the grid total is \(12x\), a multiple of 12.

Step 2: Add up every candidate number.
The four numbers already placed are 1, 15, 9 and 14. The nine free numbers to choose eight from are 2, 3, 4, 5, 7, 10, 11, 12, 13. Adding all thirteen numbers together:
\[ 1+15+9+14+2+3+4+5+7+10+11+12+13=106 \]

Step 3: Find which single number, once removed, leaves a multiple of 12.
We need \(106-(\text{dropped number})\) to be a multiple of 12. Testing the nine candidates: dropping 2 leaves 104 (not a multiple of 12); dropping 3 leaves 103 (no); dropping 4 leaves 102 (no); dropping 5 leaves 101 (no); dropping 7 leaves 99 (no); dropping 10 leaves 96 = 12 times 8 (yes); dropping 11 leaves 95 (no); dropping 12 leaves 94 (no); dropping 13 leaves 93 (no). Only dropping 10 gives a multiple of 12.

Step 4: Confirm an arrangement exists.
With 10 removed, the remaining eight numbers 2, 3, 4, 5, 7, 11, 12, 13 can indeed be arranged so every row sums to 32 and every column sums to 24 (average 8), as shown in the worked grid for question 45:

1	13	3	15
12	9	7	4
11	2	14	5

This confirms 10 is the number that must be left out, ruling out 4, 5, 7 and 11 (options 1, 2, 3 and the eliminated option 5).

Final Answer:
\[ \boxed{10} \]