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ight)^2
For \( n = 12 \), compute:
\left( rac{12 \cdot 13}{2}
ight)^2 = (78)^2 = 6084
Now compute \( 6084 \mod 13 \).
78^2 \equiv 0^2 = 0 \mod 13
Thus, the remainder when the sum of the cubes of the first 12 positive integers is divided by 13 is \( oxed{0} \).
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}