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
Step 1: Understand the problem. The problem states that when a number x is divided by 30, 45, or 54, the remainder is always 21. This means that x - 21 must be a multiple of 30, 45, and 54. To find the least value of x, we need to find the least common multiple (LCM) of 30, 45, and 54. Step 2: Find the prime factorization of each number. • For 30: 30 = 2 × 15 = 2 × 3 × 5 • For 45: 45 = 3 × 15 = 3 × 3 × 5 = 3^2 × 5 • For 54: 54 = 2 × 27 = 2 × 3 × 9 = 2 × 3 × 3 × 3 = 2 × 3^3 Step 3: Calculate the LCM of 30, 45, and 54. To find the LCM, take the highest power of each prime factor present in any of the numbers: • Highest power of 2: 2^1 • Highest power of 3: 3^3 • Highest power of 5: 5^1 LCM(30, 45, 54) = 2^1 × 3^3 × 5^1 = 2 × 27 × 5 = 270 Step 4: Determine the value of x. We know that x - 21 is the LCM. x - 21 = 270 Add 21 to both sides to find x: x = 270 + 21 x = 291 The least value of the number x is 291.