The number of terms in the expansion of $\left(5z + 7z^{-1}\right)^{1224}$ which are integers is
Step-by-Step Solution
Key Concept: Determine divisibility of binomial coefficients by analyzing the prime factorization and finding which values of $k$ satisfy the divisibility condition.
The general term coefficient is $C_k = ^{1824}C_k$. We need $1824 \mid C_k$. Since $1824 = 16 \times 114 = 2^4 \times 3 \times 19$, the coefficient $C_k$ will be divisible by $1824$ for specific values of $k$. For $\frac{1824}{K}$ to be an integer, $K = 0, 9, 18, \ldots, 1818$. Both endpoints give non-integer values for $\frac{1824}{K}$, so the divisible coefficients occur at $K = 0, 9, 18, \ldots, 1818$. The number of such terms is $102$.
Correct Answer: 102