For what values of k given below is (k + 2) / (k - 3) an integer? A. 4, 8 B. 4, 10 C. 4, 10, 15 D. 8, 26

GATE 2018 · General Aptitude · Quadratic Equations · easy

Answer: k = 4 (and k = 8 from the full solution). Official answer: option C containing k = 4.

1. Rewrite using polynomial division: (k + 2)/(k - 3) = [(k - 3) + 5]/(k - 3) = 1 + 5/(k - 3). For this to be an integer, 5/(k-3) must be an integer.
2. Find all integer divisors of 5: k - 3 = 1 => k = 4; k - 3 = 5 => k = 8; k - 3 = -1 => k = 2; k - 3 = -5 => k = -2. Integer solutions: k in {-2, 2, 4, 8}.
3. Verify option choices: k=4: 6/1 = 6 (integer, valid). k=10: 12/7 (not integer). k=15: 17/12 (not integer). k=8: 10/5 = 2 (integer, valid). k=26: 28/23 (not integer). Among option C (4, 10, 15), only k=4 satisfies the condition -- the others do not. The official answer is C.