The total number of positive divisors is \((3+1)(1+1)(1+1) = 16\). So there are 16 positive factor pairs \((a, b)\) with \(ab = 2024\), and 16 negative ones (since \(a, b\) both negative gives the same product), for a total of 32 factor pairs.

["Understanding the Number of Positive and Negative Divisors of 2024", "When exploring the divisors of a number, one key mathematical property is the total number of positive (and negative) factor pairs, which reveals deep insights into its prime factorization. Take the number 2024, a composite integer with interesting divisor structure.", "The total number of positive divisors of 2024 can be calculated by first finding its prime factorization:", "[\n2024 = 2^3 \ imes 11^1 \ imes 23^1\n]", "Using the divisor formula, if ( n = p_1^{e_1} \ imes p_2^{e_2} \ imes \cdots \ imes p_k^{e_k} ), then the number of positive divisors ( d(n) ) is:", "[\nd(n) = (e_1 + 1)(e_2 + 1) \cdots (e_k + 1)\n]", "Applying this to 2024:", "[\nd(2024) = (3 + 1)(1 + 1)(1 + 1) = 4 \ imes 2 \ imes 2 = 16\n]", "This means there are 16 positive factor pairs ((a, b)) such that ( ab = 2024 ). Each divisor ( a ) pairs uniquely with ( b = \frac{2024}{a} ). For example:", "[\n(1, 2024),\ (2, 1012),\ (4, 506),\ (8, 253),\ (11, 184),\ (22, 92),\ (44, 46)\n]", "And their corresponding negative counterparts are also valid factor pairs, since a negative times a negative is positive:", "[\n(-1, -2024),\ (-2, -1012),\ (-4, -506),\ (-8, -253),\ (-11, -184),\ (-22, -92),\ (-44, -46)\n]", "Together, the total number of factor pairs (both positive and negative) is:", "[\n16 \ ext{ (positive)} + 16 \ ext{ (negative)} = 32\n]", "This formula highlights the symmetry inherent in divisor pairs and simplifies reasoning about number structure. Knowing that 2024 has 16 positive divisors and 32 total factor pairs enables efficient computation in number theory, cryptography, and algorithm design.", "In summary, the total number of positive divisors of 2024 is ((3+1)(1+1)(1+1) = 16), leading to 16 positive factor pairs ((a, b)) and 16 negative ones, summing to 32 factor pairs in total.", "---", "Key Takeaways:", "- Positive divisors determined by prime exponent increments: ((3+1)(1+1)(1+1)).\n- Each divisor pairs uniquely with another to multiply to 2024.\n- Including negative pairs doubles the count due to ((a,b)) and ((-a,-b)).\n- Total factor pairs: 32.", "This insight makes working with divisors intuitive and systematic."]