If a and b are positive integers such that a^2 - b^2 = 21, what is the value of a?
Correct: D
a^2 - b^2 = (a + b)(a - b) = 21. Since a and b are positive integers, (a + b) and (a - b) must be integer factors of 21. The pairs of factors of 21 are (1, 21) and (3, 7). Case 1: a + b = 21 and a - b = 1. Adding the equations gives 2a = 22, so a = 11. Case 2: a + b = 7 and a - b = 3. Adding the equations gives 2a = 10, so a = 5. Since 11 is an option, it's the likely answer. To check if a=11 works: 11^2 - b^2 = 21 -> 121 - b^2 = 21 -> b^2 = 100 -> b=10. For a=5: 5^2 - b^2 = 21 -> b^2 = 4 -> b=2.