\( k \) is divisible by the sum of its digits.

February 22, 2026

$\( k \) is divisible by the sum of its digits.$

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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 \( k \)?
We seek the smallest two-digit number \( k \) such that:
\( k \equiv 2 \pmod{7} \),
Let \( k = 10a + b \), where \( a \in \{1,2,\dots,9\} \), \( b \in \{0,1,\dots,9\} \), so sum of digits is \( a + b \).
We check two-digit numbers \( k \equiv 2 \pmod{7} \): they are:
k = 16, 23, 30, 37, 44, 51, 58, 65, 72, 79, 86, 93

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