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.
ScanSolve AI Answer
Here are the solutions to your questions: Question 29: The Venn diagram shows the prime factors of x and 30. The left circle represents PF_x (Prime Factors of x). The right circle represents PF_30 (Prime Factors of 30). From the Venn diagram: Factors unique to x (left-only part of PF_x): 2^2, 3^2 Common factors (intersection): 2^1, 3^1 Factors unique to 30 (right-only part of PF_30): n a) Find the value of: i) x Step 1: To find x, multiply all the prime factors shown in the circle for PF_x. This includes the factors unique to x and the common factors. x = (2^2 × 3^2) × (2^1 × 3^1) Step 2: Combine the powers of the same prime factors. x = 2^2+1 × 3^2+1 x = 2^3 × 3^3 Step 3: Calculate the value. x = 8 × 27 x = 216 The value of x is 216. ii) n Step 1: We know that 30 = 2 × 3 × 5. From the Venn diagram, the prime factors of 30 are the common factors and the factors unique to 30. 30 = (2^1 × 3^1) × n Step 2: Simplify the common factors. 30 = (2 × 3) × n 30 = 6 × n Step 3: Solve for n. n = (30)/(6) n = 5 The value of n is 5. b) Find the G.C.F of 30 and X. Step 1: The G.C.F (Greatest Common Factor) is the product of the prime factors found in the intersection of the Venn diagram. G.C.F(30, x) = 2^1 × 3^1 Step 2: Calculate the product. G.C.F(30, x) = 2 × 3 G.C.F(30, x) = 6 The G.C.F of 30 and x is 6. c) Calculate the L.C.M of 30 and X using the Venn diagram. Step 1: The L