Mathematics

Given the ratio 2:3, let the numbers be 2x and 3x. Their LCM is 48. The LCM of 2x and 3x is 6x, so 6x = 48, giving x = 8. Therefore, the numbers are 16 and 24, and their sum is 40.

The lowest common multiple (LCM) of 24, 36, and 40 is 360. This is found by taking the highest powers of prime factors from each number: 24 = 2^3 * 3, 36 = 2^2 * 3^2, 40 = 2^3 * 5. The LCM is 2^3 * 3^2 * 5 = 8 * 9 * 5 = 360. Since 360 is divisible by 24, 36, and 40, it is the correct LCM.

The volume of a sphere is (4/3)πr³ and the volume of a cylinder is πr²h. Given equal volumes and the same radius, equate (4/3)πr³ = πr²h, which simplifies to h = 4/3 r. Thus, the height of the cylinder is 4/3 times its radius.

The side length of the square field is the square root of 69696 m², which is 264 m. The perimeter of the square is 4 × 264 = 1056 m. The wire goes around the field 15 times, so the total length required is 15 × 1056 = 15840 m, which corresponds to option A.

The volume of a cylinder is V = πr²h_cylinder, and the volume of a cone is V = (1/3)πr²h_cone. Given equal volumes and equal radii, πr²h_cylinder = (1/3)πr²h_cone, which simplifies to h_cylinder = (1/3)h_cone. Therefore, the ratio of the height of the cylinder to that of the cone is 1:3.

To find the greatest number that leaves a remainder of 3 when dividing 19, 83, and 67, subtract 3 from each number: 16, 80, and 64. The greatest number dividing all three is the GCD of 16, 80, and 64, which is 16. Hence, 16 is the required number.

The number must leave a remainder of 2 when divided by 3, 4, and 5. This means the number minus 2 is divisible by 3, 4, and 5. The least common multiple (LCM) of 3, 4, and 5 is 60, so the number minus 2 must be a multiple of 60. The smallest 4-digit number is 1000, so we find the smallest multiple of 60 greater than or equal to 998 (1000 - 2). 60 × 17 = 1020, so the number is 1020 + 2 = 1022. However, 1022 leaves remainder 2 when divided by 3 and 4, but not by 5. Checking 1026: 1026 - 2 = 1024, which is not divisible by 60. Checking 1020: 1020 - 2 = 1018, not divisible by 60. Checking 1022: 1022 - 2 = 1020, which is divisible by 60. Therefore, 1022 is the correct answer. Rechecking the calculations shows 1022 mod 5 = 1022 - (5 × 204) = 1022 - 1020 = 2, so 1022 leaves remainder 2 when divided by 5 as well. Therefore, the original correct answer D (1022) is correct. The initial explanation was missing, so this detailed explanation clarifies the reasoning.

The surface area of a cube is given by 6 times the square of its edge length. Here, the edge length is 2a, so the surface area is 6 × (2a)² = 6 × 4a² = 24a² cm². The correct option is D, which matches 24a² cm².

The product of two numbers equals the product of their H.C.F and L.C.M, i.e., 84 × 21 = 1764. Given the ratio 1:4, let the numbers be x and 4x. Then x × 4x = 1764, so 4x² = 1764, giving x² = 441 and x = 21. The larger number is 4x = 84. This is consistent since the H.C.F (21) is less than the L.C.M (84). Therefore, the larger number is 84, which corresponds to option C.

The volume of the large sphere is (4/3)π(3^3) = 36π cm³. The volume of each small sphere (shot) is (4/3)π(1^3) = (4/3)π cm³. Dividing the large volume by the small volume gives 36π ÷ (4/3)π = 36 × (3/4) = 27 shots.