Hidden Square Method in Number Theory | ISI BStat Entrance 2018 Obj 25 | Math Olympiad Preparation

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Ans: The natural numbers 'a' that satisfy the equation are a = 7 and a = 9
Explanation:
To solve (t + a - 8)(t - a + 8) = 3 for natural numbers a, we analyse the possible factorisations of 3: (1, 3), (3, 1), (-1, -3), (-3, -1).
Case 1: Set t + a - 8 = 1 and t - a + 8 = 3.
Solving these equations gives a = 7.
Case 2: Set t + a - 8 = 3 and t - a + 8 = 1.
Solving these equations gives a = 9.
Case 3: Set t + a - 8 = -1 and t - a + 8 = -3.
Solving these equations gives a = 9.
Case 4: Set t + a - 8 = -3 and t - a + 8 = -1.
Solving these equations gives a = 7.
Therefore, the natural numbers a that satisfy the equation are a = 7 and a = 9.

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