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\( k = 23 \): sum = \( 2+3 = 5 \), \( 23 \div 5 = 4.6 \), no.
\( k = 30 \): sum = \( 3+0 = 3 \), \( 30 \div 3 = 10 \), yes!
\( 67 \div 4 = 16 \times 4 = 64 \), remainder 3 → ✓
\( 67 \div 5 = 13 \times 5 = 65 \), remainder 2 → ✓
\( 67 \div 7 = 9 \times 7 = 63 \), remainder 4 → ✓
\( D^3 \equiv 1 \pmod{9} \), and
\( D \not\equiv 1 \pmod{9} \)
\( 100 \div 9 = 11 \times 9 = 99 \), remainder 1 → \( 100 \equiv 1 \pmod{9} \) → exclude
\( 101 \equiv 2 \)
\( 102 \equiv 3 \)