1. The problem involves finding a relationship between the numbers in the circular top part and the number in the rectangular bottom part of each keyhole shape. 2. For the first keyhole: numbers are 3, 2, 4 and the bottom number is 75. 3. For the second keyhole: numbers are 3, 5, 4 and the bottom number is 78. 4. Let's analyze possible operations: - Sum of top numbers for first: $3 + 2 + 4 = 9$ - Product of top numbers for first: $3 \times 2 \times 4 = 24$ - Sum of squares for first: $3^2 + 2^2 + 4^2 = 9 + 4 + 16 = 29$ 5. Check these against bottom number 75: - Try sum multiplied by something: $9 \times 8 + 3 = 75$ (but 9*8=72, +3=75, tried 8 and +3) 6. For the second keyhole: - Sum: $3 + 5 + 4 = 12$ - Check 12 × 6.5 = 78, exact match. 7. Inconsistent multiplier, try other pattern: - Check sum of squares for second: $3^2 + 5^2 + 4^2 = 9 + 25 + 16 = 50$ (not close to 78) - Product: $3 \times 5 \times 4 = 60$ 8. Consider sum of products of pairs: - For first: $(3\times2)+(2\times4)+(3\times4) = 6 + 8 + 12 = 26$, no direct match. 9. Notice difference between bottom and sum of products: - $75 - 26 = 49 = 7^2$ - $78 - 60 = 18 = 3 \times 6$ no square 10. Hypothesize bottom number = product of top three numbers + square of one number: - For first: $24 + 7^2$ no matching 7 in numbers. 11. Try sum of first two numbers multiplied by last: - For first: $(3 + 2) \times 4 = 5 \times 4 = 20$ - For second: $(3 + 5) \times 4 = 8 \times 4 = 32$ no. 12. Check difference between bottom and product: - First: $75 - 24 = 51$ - Second: $78 - 60 = 18$ - Ratio of differences: 51/18 ≈ 2.83 13. Alternative: sum of numbers squared: - $(3 + 2 + 4)^2 = 9^2 = 81$ close to 75 - $(3 + 5 + 4)^2 = 12^2 = 144$ not 78 14. Given the inconsistencies, let's assume the bottom number equals $3 \times$ (sum of the first two numbers) plus the third number squared: - First: $3 \times (3 + 2) + 4^2 = 3 \times 5 + 16 = 15 + 16 = 31$ no. 15. Since the bottom numbers 75 and 78 differ by 3 and the top numbers differ largely, we can guess the bottom number is sum of products plus sum: - First: $3\times2 + 2\times4 + 3\times4 + 3 + 2 + 4 = 6 + 8 + 12 + 9 = 35$ - No match. 16. For the bottom-left keyhole with numbers 3, 7, 5 and unknown bottom number: - Calculate product: $3 \times 7 \times 5 = 105$ - Calculate sum: $3 + 7 + 5 = 15$ - Calculate sum of squares: $9 + 49 + 25 = 83$ 17. For the bottom-right keyhole with numbers 4, 5, 3 and unknown bottom number: - Product: $4 \times 5 \times 3 = 60$ - Sum: $4 + 5 + 3 = 12$ - Sum of squares: $16 + 25 + 9 = 50$ 18. Notice first bottom number 75 is close to 3 times sum of numbers: $3 \times 9 = 27$, no. 19. Notice first bottom number 75 is close to sum of squares plus product: - $29 + 24 = 53$ no. 20. Final approach: Observe the bottom number is roughly $3 \times$ sum of top two numbers + $3 \times$ third number: - For first: $3 \times (3+2) + 3 \times 4 = 15 + 12 = 27$ no. 21. Since no direct pattern emerges easily, assume the bottom number is product of the middle number and a fixed constant plus the sum of the other two: - For first: middle number = 2, product to get 75 is $(3 + 4) + 2 \times 23 = 7 + 46 = 53$ no. 22. Without an explicit pattern, the best guess for missing numbers is the product of the three numbers: - Bottom-left missing number: $3 \times 7 \times 5 = 105$ - Bottom-right missing number: $4 \times 5 \times 3 = 60$ **Answer:** Bottom-left keyhole number is $105$. Bottom-right keyhole number is $60$.