An ornithologist uses GPS tracking to study a flock of geese whose flight paths form a repeating cycle every \( k \) days. She observes that a particular behavioral pattern occurs every \( k \) days, where \( k \) is the smallest two-digit number such that \( k \) is divisible by the sum of its digits and leaves a remainder of 2 when divided by 7. What is \( k \)?

["Understanding the Hidden Pattern: How GPS Tracking Reveals a Goose Migration Cycle", "In the intricate world of avian migration, researchers have uncovered a fascinating pattern: a flock of geese repeatedly traces a flight path that cycles every ( k ) days—a regular rhythm encoded in nature. A recent ornithological study leverages GPS tracking data to decode this cycle. By analyzing behavioral markers repeated over time, scientists discover that the smallest two-digit integer ( k ) governing this cycle satisfies two key mathematical conditions:", "1. ( k ) is divisible by the sum of its digits.\n2. ( k \equiv 2 \pmod{7} ).", "Our goal is to identify this unique two-digit number ( k )—a number that bridges natural behavior and mathematical structure.", "### Breaking Down the Problem", "We seek the smallest two-digit number ( k ) (i.e., ( 10 \leq k \leq 99 )) such that:\n- Sum of digits ( s(k) ) divides ( k ), i.e., ( k \mod s(k) = 0 ),\n- ( k \equiv 2 \pmod{7} ).", "Let’s systematically explore two-digit numbers starting from 10 upward, testing both conditions.", "#### Step 1: Generate two-digit numbers satisfying ( k \equiv 2 \pmod{7} )", "We generate all two-digit numbers congruent to 2 modulo 7:\n[\nk = 16, 23, 30, 37, 44, 51, 58, 65, 72, 79, 86, 93\n]\n(These are found by starting at ( 7 \ imes 1 + 2 = 9 ) → next is 16, and increasing by 7.)", "We now check which of these are divisible by the sum of their digits.", "#### Step 2: Test each candidate", "- ( k = 16 ): Sum = ( 1 + 6 = 7 ), ( 16 \div 7 \approx 2.285 ) → not divisible\n- ( k = 23 ): Sum = ( 2 + 3 = 5 ), ( 23 \div 5 = 4.6 ) → not divisible\n- ( k = 30 ): Sum = ( 3 + 0 = 3 ), ( 30 \div 3 = 10 ) → divisible ✅\n - Also, ( 30 \equiv 30 \mod 7 = 30 - 28 = 2 ) → satisfies ( k \equiv 2 \pmod{7} ) ✅", "So ( k = 30 ) satisfies both conditions.", "But wait—we must confirm it is the smallest such two-digit number. Let’s verify no smaller candidate meets both rules.", "The sequence starts at 16, so we’ve already ruled out 10–29. The next possible before 30 is 23 and 16, both invalid.", "Thus, 30 is the smallest two-digit number satisfying both:\n- Divisible by digit sum (3 → 30 ÷ 3 = 10),\n- Leaves remainder 2 when divided by 7.", "### Why This Matters for Bird Behavior", "The ornithologist’s GPS data reveals that the geese’s flight patterns manifest a repeating 30-day cycle—likely tied to migration efficiency, predator avoidance, or environmental cues. This mathematical regularity suggests deep biological optimization. The fact that ( k = 30 ) emerges from natural observation highlights how geometry, pattern recognition, and number theory converge in ecological research.", "Beyond academic curiosity, identifying such cycles enables better conservation planning—predicting movement patterns helps protect critical habitats during peak migration periods.", "---", "Conclusion", "From the quiet hum of wings to the precision of numbers, this study exemplifies how field biology and mathematical reasoning unite. The number ( k = 30 )—the smallest two-digit number divisible by its digit sum and congruent to 2 mod 7—governs the geese’s mysterious return. It reminds us that nature’s rhythms often follow the universe’s quiet logic: repeating, predictable, and deeply meaningful.", "Answer: ( \boxed{30} )"]