This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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12) Step 1: Find the prime factorization of 24 and 36. 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2^3 × 3^1 36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3 = 2^2 × 3^2 Step 2: Calculate the Greatest Common Divisor (GCD) of 24 and 36. The GCD is found by taking the lowest power of each common prime factor. GCD(24, 36) = 2^(3,2) × 3^(1,2) = 2^2 × 3^1 = 4 × 3 = 12 Step 3: Calculate the Least Common Multiple (LCM) of 24 and 36. The LCM is found by taking the highest power of all prime factors present. LCM(24, 36) = 2^(3,2) × 3^(1,2) = 2^3 × 3^2 = 8 × 9 = 72 Step 4: Determine the product of the LCM and GCD. Product = LCM(24, 36) × GCD(24, 36) = 72 × 12 Product = 864 Alternatively, for any two positive integers a and b, LCM(a, b) × GCD(a, b) = a × b. Product = 24 × 36 = 864 The product is C. 864. 13) Step 1: Simplify the expression inside the parentheses first. ((7)/(9) + (5)/(6) - (2)/(3)) Find a common denominator for 9, 6, and 3. The least common multiple (LCM) of 9, 6, and 3 is 18. Convert each fraction to have a denominator of 18: (7)/(9) = (7 × 2)/(9 × 2) = (14)/(18) (5)/(6) = (5 × 3)/(6 × 3) = (15)/(18) (2)/(3) = (2 × 6)/(3 × 6) = (12)/(18) Step 2: Perform the addition and subtraction inside the parentheses. (14)/(18) + (15)/(18) - (12)/(18) = (14 + 15 - 12)/(18) = (29 - 12)/(18) = (17)/(18) Step 3: Multiply the result by (1)/(3). (17)/(18) × (1)/(3) = (17 × 1)/(18 × 3) = (17)/(54) The value of the expression is D. (17)/(54). Last free one today — make it count tomorrow, or type /upgrade for unlimited.