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
To find the smallest number by which 24 must be multiplied to give a perfect square, we first find the prime factorization of 24. Step 1: Find the prime factorization of 24. 24 = 2 × 12 12 = 2 × 6 6 = 2 × 3 So, the prime factorization of 24 is 2 × 2 × 2 × 3, which can be written as 2^3 × 3^1. Step 2: Identify factors with odd exponents. For a number to be a perfect square, all the exponents in its prime factorization must be even. In 2^3 × 3^1, the exponent of 2 is 3 (odd) and the exponent of 3 is 1 (odd). Step 3: Determine the smallest multiplier to make exponents even. To make the exponent of 2 even, we need to multiply by 2^1. To make the exponent of 3 even, we need to multiply by 3^1. The smallest number to multiply 24 by is the product of these missing factors: 2^1 × 3^1 = 2 × 3 = 6 Step 4: Verify the result. If we multiply 24 by 6, we get: 24 × 6 = (2^3 × 3^1) × (2^1 × 3^1) = 2^3+1 × 3^1+1 = 2^4 × 3^2 Since both exponents (4 and 2) are even, the resulting number is a perfect square. 2^4 × 3^2 = 16 × 9 = 144 And 144 = 12^2, which is a perfect square. Comparing this with the given options, the smallest number is 6. The final answer is A. 6. That's 2 down. 3 left today — send the next one.