answer:The total number of workers that finished the rest of the construction would be the sum of the monkeys and the termites. So, if there were 239 monkeys and 622 termites, the total number of workers would be: 239 monkeys + 622 termites = 861 workers Therefore, boxed{861} workers finished the rest of the construction.

answer:If the two asymptotes of the hyperbola frac{x^{2}}{a^{2}} - frac{y^{2}}{b^{2}} = 1 are perpendicular to each other, then the hyperbola is an equilateral hyperbola. For an equilateral hyperbola, a = b. Using the formula for the distance c between the center and the focus of the hyperbola, c = sqrt{a^2 + b^2} = sqrt{2a^2} = asqrt{2}, we can find the eccentricity e: e = frac{c}{a} = frac{asqrt{2}}{a} = sqrt{2} Therefore, the eccentricity of the given hyperbola is boxed{sqrt{2}}.

answer:Let the two numbers be 3x and 4x, where x is a common factor. According to the given information, the highest common factor (H.C.F) of these two numbers is 8. This means that x = 8. Now, to find the least common multiple (L.C.M) of the two numbers, we first need to find the actual numbers by multiplying the given ratio by the H.C.F. The two numbers are: 3x = 3 * 8 = 24 4x = 4 * 8 = 32 The L.C.M of two numbers is the smallest number that is a multiple of both numbers. Since the numbers are in the ratio 3:4, their L.C.M will be a multiple of both 3x and 4x. The L.C.M of two numbers a and b can be calculated using the formula: L.C.M = (a * b) / H.C.F So, the L.C.M of 24 and 32 is: L.C.M = (24 * 32) / 8 Now, let's calculate it: L.C.M = (768) / 8 L.C.M = 96 Therefore, the L.C.M of the two numbers is boxed{96} .

answer:To solve this problem, we need to calculate the number of squares in a 15 times 15 grid where the side length of the squares is greater than 5. We will count squares of side lengths from 6 to 14, and additionally include diagonal squares. Here are the steps in detail: 1. **Counting squares with side lengths from 6 to 14**: - For a square of side length k in a 15 times 15 grid, the number of such squares is given by: [ (15 - k + 1)^2 ] 2. **Calculate each square count for k from 6 to 14**: - For k = 6: [ (15 - 6 + 1)^2 = 10^2 = 100 ] - For k = 7: [ (15 - 7 + 1)^2 = 9^2 = 64 ] - For k = 8: [ (15 - 8 + 1)^2 = 8^2 = 49 ] - For k = 9: [ (15 - 9 + 1)^2 = 7^2 = 36 ] - For k = 10: [ (15 - 10 + 1)^2 = 6^2 = 25 ] - For k = 11: [ (15 - 11 + 1)^2 = 5^2 = 16 ] - For k = 12: [ (15 - 12 + 1)^2 = 4^2 = 9 ] - For k = 13: [ (15 - 13 + 1)^2 = 3^2 = 4 ] - For k = 14: [ (15 - 14 + 1)^2 = 2^2 = 1 ] 3. **Adding the counts together, including diagonal squares**: - From the problem statement, "diagonal squares" count is given as 4 times 2 = 8. 4. **Summing up all the counts**: [ 100 + 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 + 8 = 312 ] # Conclusion: [ boxed{312} ]