Mathematics

The least number of five digits divisible by 12, 15, and 18 is the least common multiple (LCM) of these numbers that is at least 10000. The LCM of 12, 15, and 18 is 180. The smallest five-digit multiple of 180 is 10080 (180 × 56 = 10080).

The base area is given as 6500 square centimeters. Converting to square meters: 6500 cm² = 6500 × 0.0001 m² = 0.65 m². The volume of water is given as 2.6 cubic meters. The depth of water is calculated by dividing the volume by the base area: depth = volume / base area = 2.6 m³ / 0.65 m² = 4 m. Therefore, the correct answer is D (4 m). The original correct_answer was B, which is incorrect based on the calculation and unit conversion.

The least common multiple (LCM) of 455, 117, and 338 is 106470. This is found by prime factorizing each number and taking the highest powers of all prime factors involved, then multiplying them together.

The least number which when diminished by 7 is divisible by 21, 28, 36, and 45 means the number minus 7 is a common multiple of these numbers. The least common multiple (LCM) of 21, 28, 36, and 45 is 1260. Therefore, the number is 1260 + 7 = 1267. However, option A is 1267, which matches this calculation, so the original correct answer A is correct. But the question asks for the least number which when diminished by 7 is divisible by those numbers, so the number minus 7 must be divisible by all. Since 1260 is divisible by all, the number is 1267. So the correct answer is A.

Let the four primes be p1 < p2 < p3 < p4. Given p1 * p2 * p3 = 7429 and p2 * p3 * p4 = 12673. Dividing the second product by the first gives p4 / p1 = 12673 / 7429 = 29 / 17 exactly. Since p1 and p4 are primes, p1 = 17 and p4 = 29. Factoring 7429 confirms it is 17 * 19 * 23, and factoring 12673 confirms it is 19 * 23 * 29. Therefore, the four primes are 17, 19, 23, and 29, and the fourth number is 29. Hence, the correct answer is B.

The circumference of a circle is directly proportional to its radius (C = 2πr). If the radius is increased 3 times, the circumference also increases 3 times.

The largest bamboo that can fit inside a square of area 100 sq. meters corresponds to the square's diagonal length. The side length of the square is √100 = 10 meters, so the diagonal is 10√2 ≈ 14.14 meters, which is option B.

The square field has an area of 12100 m², so its side length is √12100 = 110 m. The perimeter of the square is 4 × 110 = 440 m. The circle has the same perimeter (circumference) of 440 m, so 2πr = 440 ⇒ r = 440/(2π) = 70 m. The area of the circle is πr² = π × 70² ≈ 15393 m², which is closest to option D (15300 Sq.m).

When the length is increased by 20%, it becomes 1.2 times the original length. When the breadth is decreased by 20%, it becomes 0.8 times the original breadth. The new area is 1.2 × 0.8 = 0.96 times the original area, which is a 4% decrease.

The wheels revolve at 15, 20, and 48 revolutions per minute (rpm). To find when they will all align at the same position starting from a certain point, we need to find the time interval after which their angular positions coincide again. The key is to find the least common multiple (LCM) of their revolution periods (time per revolution), not the LCM of their rpm values. The period for each wheel is the reciprocal of its rpm: Wheel 1 period = 1/15 min, Wheel 2 period = 1/20 min, Wheel 3 period = 1/48 min. The LCM of these periods corresponds to the greatest common divisor (GCD) of their rpm values. The GCD of 15, 20, and 48 is 1. Therefore, the wheels will align every 1 minute. Hence, the correct answer is 1 minute (option A).