Mathematics

The product of the H.C.F and L.C.M of two numbers equals the product of the numbers themselves. Since the H.C.F is 8, the L.C.M must be a multiple of 8. Among the options, 60 is not divisible by 8, so it can never be their L.C.M.

The volume of the metal sheet is 27 × 8 × 1 = 216 cm³. When melted into a cube, the cube's side length is the cube root of 216, which is 6 cm. The surface area of the original cuboid is 2(27×8 + 8×1 + 27×1) = 2(216 + 8 + 27) = 2×251 = 502 cm². The surface area of the cube is 6 × 6² = 6 × 36 = 216 cm². The difference is 502 - 216 = 286 cm², which corresponds to option B.

The problem requires finding the smallest number such that when increased by 5, the result is divisible by 6, 9, 15, 20, and 24. This means (number + 5) must be a common multiple of these numbers. The least common multiple (LCM) of 6, 9, 15, 20, and 24 is 360. Therefore, number + 5 = 360, so the number = 360 - 5 = 355. Among the options, 355 corresponds to option B, making it the correct answer.

If the sides of two cubes are in the ratio 1:2, their volumes are in the ratio of the cubes of their sides, which is 1³:2³ = 1:8. Hence, the volume ratio is 1:8.

The volume of a cylinder is given by V = πr²h, and the volume of a cone is V = (1/3)πr²h. Since both have the same radius and height, the ratio of their volumes is cylinder : cone = πr²h : (1/3)πr²h = 3 : 1.

The least common multiple (LCM) of 3, 4, and 5 is 60. The greatest 4-digit number is 9999. The largest multiple of 60 less than or equal to 9999 is 9975 (60 × 166). Therefore, 9975 is the greatest 4-digit number divisible by 3, 4, and 5.

The volume of the original ball is equal to the sum of the volumes of the three smaller balls. Using the formula for the volume of a sphere (V = 4/3 π r³), we calculate the volume of the original ball and subtract the volumes of the two smaller balls to find the volume of the third ball. From this, we find the diameter of the third ball to be approximately 2.66 cm.

The volumes of cubes are proportional to the cubes of their side lengths. Given the volume ratio 27:125, the side length ratio is the cube root, 3:5. Surface area is proportional to the square of the side length, so the surface area ratio is 3²:5² = 9:25.

The volume of the cube is 1728 cm³. The side length is the cube root of 1728, which is 12 cm. The surface area of a cube is 6 times the square of the side length: 6 × 12² = 6 × 144 = 864 cm². However, the options given are in cm2 but option A is 864 cm2, which matches the calculation. Therefore, the correct answer is A, not B. The initial correct_answer was A, which is correct. The explanation was missing and has been added. The difficulty is easy based on the calculation required. The tags have been added accordingly.

First, calculate each number: 4*27*3125 = 337500, 8*9*25*7 = 12600, and 16*81*5*11*49 = 3499200. The HCF is found by taking the common prime factors with the smallest powers. The common prime factors are 2, 3, and 5. The minimum powers are 2^2 (4), 3^2 (9), and 5^1 (5). Multiplying these gives 4*9*5 = 180.