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To determine how many factors of 14,400 (which can be expressed as 2^6 × 3^2 × 5^2 × 7^1) are divisible by 18 but not by 36, we need to consider the prime factorization. Factors divisible by 18 (2 × 3^2) must include at least two factors of 3 and one factor of 2. To exclude those divisible by 36 (2^2 × 3^2), the factor must not include two factors of 2. Using combinatorics, we calculate the valid combinations of factors. Similarly, for 2^3×3^3×5^4×7^2 and factors divisible by 50 (2 × 5^2) but not 100 (2^2 × 5^2), we exclude those with two factors of 2.

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