To solve the problem, we need to analyze the equation given for three consecutive positive integers \( k \), \( l \), and \( m \). The relationship provided is: \[ l^2 = 3(k + m) \] ### Step 1: Define the integers Since \( k \), \( l \), and \( m \) are consecutive integers, we can express them in terms of \( l \): - Let \( l = n \) (the middle integer), - Then \( k = n - 1 \) (the integer before), - And \( m = n + 1 \) (the integer after). ### Step 2: Substitute into the equation Now, substituting \( k \) and \( m \) into the equation: \[ l^2 = 3(k + m) \implies n^2 = 3((n - 1) + (n + 1)) \] ### Step 3: Simplify the right side The expression inside the parentheses simplifies as follows: \[ (n - 1) + (n + 1) = n - 1 + n + 1 = 2n \] Thus, we can rewrite the equation as: \[ n^2 = 3(2n) \] ### Step 4: Solve for \( n \) Now, we simplify the equation: \[ n^2 = 6n \] Rearranging gives us: \[ n^2 - 6n = 0 \] Factoring out \( n \): \[ n(n - 6) = 0 \] This gives us two solutions: 1. \( n = 0 \) (not a positive integer) 2. \( n = 6 \) (valid solution) ### Step 5: Find \( k \), \( l \), and \( m \) Since \( n = 6 \): - \( l = n = 6 \) - \( k = n - 1 = 5 \) - \( m = n + 1 = 7 \) ### Step 6: Conclusion The value of \( m \) is therefore: \[ \boxed{7} \] ### Explanation of Other Options - **Option A (4)**: If \( m = 4 \), then \( l \) would have to be 3 and \( k \) would be 2. Substituting these values into the equation \( l^2 = 3(k + m) \) gives \( 3^2 = 3(2 + 4) \) or \( 9 = 18 \), which is false. - **Option B (5)**: If \( m = 5 \), then \( l = 4 \) and \( k = 3 \). Substituting gives \( 4^2 = 3(3 + 5) \) or \( 16 = 24 \), which is also false. - **Option C (6)**: If \( m = 6 \), then \( l = 5 \) and \( k = 4 \). Substituting gives \( 5^2 = 3(4 + 6) \) or \( 25 = 30 \), which is false. ### Revision Summary - The integers \( k \), \( l \), and \( m \) are consecutive integers. - The equation \( l^2 = 3(k + m) \) can be simplified by expressing \( k \) and \( m \) in terms of \( l \). - The valid solution for \( l \) is 6, leading to \( m = 7 \). - Other options do not satisfy the equation when substituted back.