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
You're on a roll — here are the solutions to questions 9, 10, and 11: Question 9: The GCD of three numbers is 30 and their LCM is 900. Two of the numbers are 60 and 150. We need to find the least possible value of the third number. Let the three numbers be A, B, C. Given: A = 60, B = 150. GCD(A, B, C) = 30 LCM(A, B, C) = 900 Step 1: Express the given numbers, GCD, and LCM as products of their prime factors. 30 = 2^1 × 3^1 × 5^1 900 = 9 × 100 = 3^2 × 10^2 = 3^2 × (2 × 5)^2 = 2^2 × 3^2 × 5^2 A = 60 = 2^2 × 3^1 × 5^1 B = 150 = 2^1 × 3^1 × 5^2 Let the third number be C = 2^x × 3^y × 5^z. Step 2: Use the properties of GCD to determine the minimum exponents. For GCD, the exponent of each prime factor is the minimum of its exponents in the numbers. GCD(A, B, C) = 2^(2, 1, x) × 3^(1, 1, y) × 5^(1, 2, z) = 2^1 × 3^1 × 5^1 This implies: (2, 1, x) = 1 x 1 (1, 1, y) = 1 y 1 (1, 2, z) = 1 z 1 Step 3: Use the properties of LCM to determine the maximum exponents. For LCM, the exponent of each prime factor is the maximum of its exponents in the numbers. LCM(A, B, C) = 2^(2, 1, x) × 3^(1, 1, y) × 5^(1, 2, z) = 2^2 × 3^2 × 5^2 This implies: (2, 1, x) = 2 x 2 (1, 1, y) = 2 y = 2 (since y 1) (1, 2, z) = 2 z = 2 (since z 1) Step 4: Determine the least possible value for C. To find the least possible value of C, we choose the smallest possible exponents for x, y, z that satisfy all conditions. For x: x 1 and x 2. The smallest possible integer value is x=1. For y: y=2. For z: z=2. So, C = 2^1 × 3^2 × 5^2. C = 2 × 9 × 25 C = 18 × 25 C = 450 The least possible value of the third number is 450. Question 10: Solve for x in the equation 27^x × 9^2x((1)/(3))^x = 81. Step 1: Express all terms with a common base, which is 3. 27 = 3^3 9 = 3^2 (1)/(3) = 3^-1 81 = 3^4 Substitute these into the equation: (3^3)^x × (3^2)^2x(3^-1)^x = 3^4 Step 2: Apply the power rule (a^m)^n = a^mn. 3^3x × 3^4x3^-x = 3^4 Step 3: Apply the product rule a^m × a^n = a^m+n in the numerator. 3^3x+4x3^-x = 3^4 3^7x3^-x = 3^4 Step 4: Apply the quotient rule (a^m)/(a^n) = a^m-n. 3^7x - (-x) = 3^4 3^7x+x = 3^4 3^8x = 3^4 Step 5: Equate the exponents since the bases are the same. 8x = 4 Step 6: Solve for x. x = (4)/(8) x = (1)/(2) The value of x is (1)/(2). Question 11: The sum of interior angles of two regular polygons of sides n and n+2 are in the ratio 3:4. Calculate the value of n and hence find the size of each exterior angle of the polygon with n sides. Step 1: Write the formula for the sum of interior angles of a polygon. The sum of interior angles of a polygon with k sides is S_k = (k-2) × 180^. Step 2: Set up the ratio for the sums of interior angles. For the polygon with n sides: S_n = (n-2) × 180^. For the polygon with n+2 sides: S_n+2 = ((n+2)-2) × 180^ = n × 180^. The ratio S_n : S_n+2 = 3:4 means: ((n-2) × 180^)/(n × 180^) = (3)/(4) Step 3: Solve for n. Cancel out 180^ from the numerator and denominator: (n-2)/(n) = (3)/(4) Cross-multiply: 4(n-2) = 3n 4n - 8 = 3n 4n - 3n = 8 n = 8 The value of n is 8. Step 4: Find the size of each exterior angle of the polygon with n sides. The polygon has n=8 sides. The formula for each exterior angle of a regular polygon with k sides is E_k = (360^)/(k). For the polygon with n=8 sides: Exterior Angle = (360^)/(8) Exterior Angle = 45^ The size of each exterior angle of the polygon with n sides is 45^. What's next?