Mathematics

The highest common factor (H.C.F) of 12, 16, 18, and 24 is 2 because 2 is the largest number that divides all four numbers exactly.

The rectangle's perimeter is calculated as 2 × (18 cm + 26 cm) = 88 cm. Since the circle has the same perimeter, its circumference is also 88 cm. Using the circumference formula C = 2πr, we solve for the radius r = 88 / (2π) = 44 / π. Using π ≈ 22/7, r = 44 × 7 / 22 = 14 cm. The area of the circle is then A = πr² = (22/7) × 14 × 14 = 616 cm². Therefore, the correct answer is option C (616 cm²).

The number of divisors for each option is as follows: 99 has 6 divisors (1, 3, 9, 11, 33, 99), 101 is prime with 2 divisors (1, 101), 176 has 10 divisors (1, 2, 4, 8, 11, 16, 22, 44, 88, 176), and 182 has 8 divisors (1, 2, 7, 13, 14, 26, 91, 182). Therefore, 176 has the greatest number of divisors among the given numbers.

Given the ratio 2:3, let the numbers be 2x and 3x. Their LCM is 48, which equals 6x (since LCM of 2x and 3x is 6x). So, 6x = 48 implies x = 8. Therefore, the numbers are 16 and 24, and their sum is 40. However, the sum 40 is option C, but the LCM calculation shows 6x = 48, so x=8, numbers 16 and 24, sum 40. So the correct answer is C, not B. Rechecking options and explanation: The original correct answer C (40) is correct. The explanation was missing and is now added.

To find the least five-digit number divisible by 144, we divide 10000 by 144, which gives approximately 69.44. Multiplying 144 by 70 gives 10080, which is the smallest five-digit number divisible by 144. Hence, the least number of five digits with HCF 144 is 10080.

The highest common factor (H.C.F) of 204, 1190, and 1445 is 17 because 17 is the largest number that divides all three numbers exactly.

The volume of water needed to raise the level by 20 cm in the tank is calculated by multiplying the base area by the height increase. Then, dividing this volume by the aperture area and the time gives the required flow speed. The correct calculation leads to approximately 5200 m/hr, matching option B.

Each man completes one lap of 11 km in times: 11/4 = 2.75 hours, 11/5 = 2.2 hours, and 11/8 = 1.375 hours respectively. To find when they all meet again at the starting point, we need the least common multiple (LCM) of these lap times. Instead of directly finding the LCM of the fractional times, consider the number of laps each completes in a given time. They will meet when each has completed an integer number of laps simultaneously. The speeds are 4, 5, and 8 km/h, so after 11 hours, the first man completes 4 × 11 = 44 laps, the second 5 × 11 = 55 laps, and the third 8 × 11 = 88 laps. Since 11 hours is the smallest time when all have completed an integer number of laps, they meet again at the starting point after 11 hours. Therefore, the correct answer is 11 hours (option A).

The product of the two numbers is 2160 and their HCF is 12. Let the numbers be 12x and 12y, where x and y are co-prime. Then, 12x * 12y = 2160 implies 144xy = 2160, so xy = 15. The pairs of co-prime numbers with product 15 are (3,5). Thus, the numbers are (12*3, 12*5) = (36, 60), which matches option A.

The product of two numbers equals the product of their HCF and LCM. Given the product is 2025 and HCF is 15, LCM = 2025 ÷ 15 = 135. However, 135 is option C, so the correct answer is C, not D. Rechecking: 15 × 135 = 2025, which matches the product. Therefore, the correct answer is C.