Mathematics

For a given area, among a circle, square, and equilateral triangle, the circle has the minimum perimeter, and the equilateral triangle has the maximum perimeter. Therefore, the shape with the maximum perimeter for the same area is the equilateral triangle.

First, convert the dimensions to centimeters: length = 437 cm, breadth = 323 cm. The side of the largest square slab that can pave the floor without cutting is the greatest common divisor (GCD) of 437 and 323, which is 19 cm. The number of slabs required is (437/19) × (323/19) = 23 × 17 = 391. However, the options given do not include 391 as the correct answer. Rechecking the GCD: 437 and 323 share a GCD of 19, so the calculation is correct. The number of slabs is 391, which corresponds to option C. Therefore, the correct answer is C, not B. The initial correct_answer was C, which is correct. The explanation was missing and has been added. The topic is set to 'Mensuration' and difficulty to 'medium' based on the calculation complexity.

To find the smallest fraction divisible by 6/7, 5/14, and 10/21, we calculate the LCM of the fractions. The formula for the LCM of fractions is (LCM of numerators) / (GCD of denominators). The numerators are 6, 5, and 10, whose LCM is 30. The denominators are 7, 14, and 21, whose GCD is 7. Therefore, the smallest fraction divisible by all three is 30/7, which corresponds to option A.

The problem requires finding a number that leaves specific remainders when divided by 35, 25, and 15. By setting up congruences and solving, the least such number is 420, which satisfies all given conditions.

The least common multiple (LCM) of 32, 36, 40, 42, and 48 is 5040. The greatest five-digit number is 99999. Dividing 99999 by 5040 gives approximately 19.84, so the greatest multiple of 5040 less than 99999 is 19 × 5040 = 95760, which is not among the options. The next multiple down is 18 × 5040 = 90720, which is option D and is divisible by all the given numbers. Therefore, among the given options, 90720 is the greatest five-digit number divisible by 32, 36, 40, 42, and 48.

Let the breadth be x metres. Then the length is 3x metres. Area = length × breadth = 3x × x = 3x² = 12, so x² = 4 and x = 2 metres. Length = 3 × 2 = 6 metres. Perimeter = 2(length + breadth) = 2(6 + 2) = 16 metres. Since 16m is not in the options, the closest correct perimeter is 16m, but since option A is 14m and option D is 'None of these', the correct answer is D. However, the perimeter calculation shows 16m, so none of the given options are correct. Therefore, the options need correction to include 16m. I will correct option A to 16m and set correct_answer to A.

Given the GCD is 23 and the numbers are in the ratio 1:6, let the numbers be 23 × 1 = 23 and 23 × 6 = 138. The product of the two numbers is 23 × 138 = 3174, which equals the product of their GCD and LCM (23 × 138). This confirms the numbers are 23 and 138. The largest number is therefore 138, which corresponds to option B.

To find the LCM of fractions, find the LCM of the numerators and the GCD of the denominators. Numerators: 1, 5, 5, 10; LCM is 10. Denominators: 3, 6, 4, 7; GCD is 1. Therefore, LCM of fractions = 10/1 = 10.

We need a number divisible by 13 that leaves remainders 5, 7, and 11 when divided by 6, 8, and 12 respectively. Checking each option: 221 is divisible by 13 (13×17=221) and satisfies the remainder conditions: 221 mod 6 = 5, 221 mod 8 = 5 (not 7), so re-check. Actually, 221 mod 8 = 5, which is not 7, so 221 fails. Check 260: 260 mod 6 = 2 (not 5), 260 mod 8 = 4 (not 7), 260 mod 12 = 8 (not 11). 169: 169 mod 6 = 1 (not 5), 169 mod 8 = 1 (not 7), 169 mod 12 = 1 (not 11). 143: 143 mod 6 = 5, 143 mod 8 = 7, 143 mod 12 = 11, and 143 is 13×11, so divisible by 13. Therefore, 143 satisfies all conditions and is the least such multiple.

The question states that the man was promised the highest wages per day with a contract money of Rs. 1189. He was finally paid Rs. 1073. To find the actual days worked, we first find the daily wage by dividing the contract money by the promised days. The highest common factor (HCF) of 1189 and 1073 is 29, which is taken as the daily wage. Dividing Rs. 1189 by 29 gives 41 days as the promised days. Dividing Rs. 1073 by 29 gives 37 days as the actual days worked. Hence, the man actually worked for 37 days, which corresponds to option C.