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
Right alishabayambumagar, let's go. Here are the solutions for question 7: 7. 36, 45 and 54 are three given numbers. i) Find the H.C.F. of these numbers. Step 1: Find the prime factorization of each number. 36 = 2 × 2 × 3 × 3 = 2^2 × 3^2 45 = 3 × 3 × 5 = 3^2 × 5^1 54 = 2 × 3 × 3 × 3 = 2^1 × 3^3 Step 2: Identify the common prime factors and their lowest powers. The common prime factor is 3. The lowest power of 3 present in all factorizations is 3^2. Step 3: Multiply the common prime factors with their lowest powers to find the H.C.F. H.C.F. = 3^2 = 3 × 3 = 9 The H.C.F. of 36, 45, and 54 is 9. ii) Find the greatest number that can be added to 2 so that the sum divides these numbers exactly. Step 1: Understand the problem. We are looking for a number, let's call it X. When X is added to 2, the sum (X+2) must divide 36, 45, and 54 exactly. This means (X+2) is a common factor of these numbers. Step 2: To find the greatest such number X, the sum (X+2) must be the greatest common factor (H.C.F.) of 36, 45, and 54. Step 3: From part (i), we found that the H.C.F. of 36, 45, and 54 is 9. So, we set the sum equal to the H.C.F.: X + 2 = 9 Step 4: Solve for X. X = 9 - 2 X = 7 The greatest number that can be added to 2 is 7. What's next?