k \equiv 3 \pmod{5} \

February 22, 2026

$k \equiv 3 \pmod{5} \$

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Question: A cyber-resilience consultant in Toronto is modeling secure key rotation intervals and observes that a system updates its encryption key every \( k \) hours, where \( k \) is the smallest positive integer such that \( k \equiv 3 \pmod{5} \), \( k \equiv 4 \pmod{7} \), and \( k \equiv 2 \pmod{9} \). What is the value of \( k \)?
Solution: We solve the system of congruences:
egin{cases}
k \equiv 4 \pmod{7} \
k \equiv 2 \pmod{9}
We apply the Chinese Remainder Theorem step by step.

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