To solve the problem, we need to find three consecutive positive integers \( k \), \( l \), and \( m \) such that the equation \( l^2 = 3(k + m) \) holds true. Let's break this down step-by-step. ### Step 1: Define the Consecutive Integers Since \( k \), \( l \), and \( m \) are consecutive integers, we can express them in terms of \( l \): - Let \( k = l - 1 \) - Let \( l = l \) - Let \( m = l + 1 \) ### Step 2: Substitute into the Equation Now, we substitute \( k \) and \( m \) into the equation \( l^2 = 3(k + m) \): \[ l^2 = 3((l - 1) + (l + 1)) \] This simplifies to: \[ l^2 = 3(2l) \] \[ l^2 = 6l \] ### Step 3: Rearrange the Equation Next, we rearrange the equation to bring all terms to one side: \[ l^2 - 6l = 0 \] ### Step 4: Factor the Quadratic We can factor the left-hand side: \[ l(l - 6) = 0 \] This gives us two potential solutions: 1. \( l = 0 \) 2. \( l = 6 \) Since we are looking for positive integers, we discard \( l = 0 \) and keep \( l = 6 \). ### Step 5: Find \( k \) and \( m \) Now that we have \( l = 6 \), we can find \( k \) and \( m \): - \( k = l - 1 = 6 - 1 = 5 \) - \( m = l + 1 = 6 + 1 = 7 \) ### Step 6: Verify the Solution We need to verify that these values satisfy the original equation: \[ l^2 = 3(k + m) \] Substituting \( k = 5 \), \( l = 6 \), and \( m = 7 \): \[ 6^2 = 3(5 + 7) \] Calculating both sides: \[ 36 = 3(12) \] \[ 36 = 36 \] The equation holds true, confirming our values are correct. ### Conclusion The value of \( m \) is \( 7 \). ### Explanation of Other Options - **Option A (4)**: If \( m = 4 \), then \( l \) would have to be \( 3 \) (since \( m = l + 1 \)), leading to \( k = 2 \). Substituting these values into the equation does not satisfy \( l^2 = 3(k + m) \). - **Option B (5)**: If \( m = 5 \), then \( l \) would have to be \( 4 \), leading to \( k = 3 \). Again, substituting these values does not satisfy the equation. - **Option C (6)**: If \( m = 6 \), then \( l \) would have to be \( 5 \), leading to \( k = 4 \). This does not satisfy the equation either. ### Revision Summary - Consecutive integers can be expressed in terms of one variable. - Substitute and simplify the equation to find relationships between the integers. - Factor the resulting quadratic equation to find possible values. - Verify the solution by substituting back into the original equation. The correct answer is **D. 7**.