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
Alright Ela — let's do this. 1. Find the LCM of the given numbers. i) 12 and 30 Step 1: Find the prime factorization of each number. 12 = 2^2 × 3 30 = 2 × 3 × 5 Step 2: Multiply the highest powers of all prime factors. LCM(12, 30) = 2^2 × 3 × 5 = 4 × 3 × 5 = 60 ii) 16 and 50 Step 1: Find the prime factorization of each number. 16 = 2^4 50 = 2 × 5^2 Step 2: Multiply the highest powers of all prime factors. LCM(16, 50) = 2^4 × 5^2 = 16 × 25 = 400 iii) 18, 20 and 36 Step 1: Find the prime factorization of each number. 18 = 2 × 3^2 20 = 2^2 × 5 36 = 2^2 × 3^2 Step 2: Multiply the highest powers of all prime factors. LCM(18, 20, 36) = 2^2 × 3^2 × 5 = 4 × 9 × 5 = 180 iv) 55, 70 and 77 Step 1: Find the prime factorization of each number. 55 = 5 × 11 70 = 2 × 5 × 7 77 = 7 × 11 Step 2: Multiply the highest powers of all prime factors. LCM(55, 70, 77) = 2 × 5 × 7 × 11 = 770 v) 45, 60, 90 and 120 Step 1: Find the prime factorization of each number. 45 = 3^2 × 5 60 = 2^2 × 3 × 5 90 = 2 × 3^2 × 5 120 = 2^3 × 3 × 5 Step 2: Multiply the highest powers of all prime factors. LCM(45, 60, 90, 120) = 2^3 × 3^2 × 5 = 8 × 9 × 5 = 360 2. Find the smallest number which on adding 11 becomes exactly divisible by 24, 36 and 60. Step 1: Let the required number be N. If N+11 is exactly divisible by 24, 36, and 60, then N+11 must be the LCM of these numbers. Step 2: Find the LCM of 24, 36, and 60. 24 = 2^3 × 3 36 = 2^2 × 3^2 60 = 2^2 × 3 × 5 LCM(24, 36, 60) = 2^3 × 3^2 × 5 = 8 × 9 × 5 = 360 Step 3: Set up the equation and solve for N. N + 11 = 360 N = 360 - 11 N = 349 3. Find the smallest number which leaves a remainder 7 when divided by 20, 50 and 75. Step 1: Let the required number be N. If N leaves a remainder 7 when divided by 20, 50, and 75, then N-7 must be exactly divisible by 20, 50, and 75. Step 2: Find the LCM of 20, 50, and 75. 20 = 2^2 × 5 50 = 2 × 5^2 75 = 3 × 5^2 LCM(20, 50, 75) = 2^2 × 3 × 5^2 = 4 × 3 × 25 = 300 Step 3: Set up the equation and solve for N. N - 7 = 300 N = 300 + 7 N = 307 Send me the next one 📸