Three positive integers a, b, with > satisfy the following equations: ac + + = bc + + 66, + + = 32. Find the value of a.
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. A, b, I+.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. ac + + = bc + + 66, + + = 32.
Hint 3: Proceed with the final algebraic steps to solve the system. => c(a – b) + (b – a) + solve for the final value.
Step 1: A, b, I+
Step 2: ac + + = bc + + 66, + + = 32
Step 3: => c(a – b) + (b – a) + = 66
Step 4: => (a – b) (c – 1) + (c – 1) = 66 – 1
Step 5: => (c – 1) (a – + 1) = 65 = 1 × 65 = 65 × 1
Step 6: = 5 × 13 = 13 × 5
Step 7: c – 1 = 65, – + 1 = 1
Step 8: => = 66, but + + = 32
Step 9: => not possible
Step 10: c – 1= 1, – + 1 = 65
Step 11: c = 2, – = 64
Step 12: a + + = 32 => + = 30
Step 13: => 2a = 64 => = 47, = –17 not possible
Step 14: c – 1 = 5, – + 1 = 13
Step 15: c = 6, + i = 26
Step 16: a – = 12
Step 17: => = 19, = 7
Step 18: c – 1 = 13, – + 1 = 5
Step 19: => – 1 = 13, – + 1 = 5
Step 20: => = 14, – = 4
Step 21: a = = 18
Step 22: => = 11 but < c
Step 23: => only = 19
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