Mathematics

Given the area A = 24.64 m², the radius r is calculated using the formula A = πr². Thus, r = √(24.64/π) ≈ √(7.84) ≈ 2.8 m. The circumference C is then C = 2πr ≈ 2 × 3.14 × 2.8 ≈ 17.6 m. Therefore, the correct answer is option C (17.60 m). Option B (16.36 m) is incorrect and option A has wrong units (cm instead of m).

The number of stones leaves a remainder of 0 when divided by 21, and a remainder of 3 when divided by 16, 20, 25, and 45. This means the number minus 3 is divisible by 16, 20, 25, and 45. The least common multiple (LCM) of these four numbers is 7200, so the smallest number satisfying the conditions is 7200 + 3 = 7203.

The least common multiple (LCM) of 5, 6, 4, and 3 is 60. To find the least number to add to 2497 so that the sum is divisible by 60, we calculate 2497 mod 60, which is 37. The difference between 60 and 37 is 23. Therefore, adding 23 to 2497 results in 2520, which is divisible by 5, 6, 4, and 3.

When the side length of a square is increased by 25%, the new side length is 1.25 times the original. The area increases by (1.25)^2 - 1 = 1.5625 - 1 = 0.5625, or 56.25%.

Given the perimeter P = 60m and length L = 2 × breadth B, we have 2(L + B) = 60, so L + B = 30. Substituting L = 2B gives 2B + B = 30, so 3B = 30 and B = 10m. Then L = 20m. Area = L × B = 20 × 10 = 200 m².

Given both shapes have the same perimeter of 160m, the square has sides of 40m (since 4 × 40 = 160). The rectangle's area is 100 sq.m less than the square's area. Using the perimeter and area difference, the rectangle's length calculates to 40m.

The least number divisible by 15, 25, 40, and 75 is the least common multiple (LCM) of these numbers. The LCM is 600. The smallest four-digit number divisible by 600 is 9000 (600 × 15).

To find the greatest number that leaves remainders 6 and 5 when dividing 1657 and 2037 respectively, subtract the remainders from the numbers: 1657 - 6 = 1651 and 2037 - 5 = 2032. The greatest number dividing both 1651 and 2032 exactly is their greatest common divisor (GCD), which is 123.

The question asks for the greatest four-digit number whose highest common factor (HCF) with 144 is 144 itself. This means the number must be divisible by 144. The largest four-digit number is 9999. Dividing 9999 by 144 gives approximately 69.43, so the greatest multiple of 144 less than 10000 is 144 × 69 = 9936. Among the options given, 9936 is divisible by 144 and is the greatest number satisfying the condition. Therefore, the correct answer is D (9936).

Let the radius of the smaller circle be r cm, and the bigger circle be r + 7 cm. The difference in areas is π[(r + 7)^2 - r^2] = 1078. Simplifying gives π(14r + 49) = 1078, so 14r + 49 = 1078/π ≈ 343. Solving for r gives r ≈ 17.5 cm.