Problem 12 · 2024 Math Kangaroo
Medium
Number Theory
factor-pairs
Four different positive whole numbers are written into the grid and then covered up. The product of the two numbers in each row, and in each column, is written next to or below the grid (see diagram). What is the sum of the four covered numbers?
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Answer: C — 13
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Hint 1 of 2
Label the four cells and write the four product equations for the rows and columns.
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Hint 2 of 2
All four numbers are different; use the column products 4 and 12 with the row products 6 and 8 to pin them down.
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Approach: solve the product equations for four distinct integers
- Let the top row be a, b and bottom row c, d.
- Rows: a·b = 6, c·d = 8. Columns: a·c = 4, b·d = 12.
- Trying a = 1 gives c = 4, b = 6, d = 2, all different: numbers 1, 6, 4, 2.
- Their sum is 1 + 6 + 4 + 2 = 13.
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