The number of positive divisors is $ (4+1)(2+1) = 15 $, so there are 15 positive divisor pairs $ (a, b) $ with $ a > 0, b > 0 $, and another 15 with $ a < 0, b < 0 $, giving 30 total integer divisor pairs (since $ a $ and $ b $ can both be negative).

["Understanding Positive Divisors, Their Divisor Count, and Integer Divisor Pairs", "When exploring the properties of positive integers—especially their divisors—mathematicians often calculate the number of positive divisors using prime factorization. One interesting and elegant formula arises when analyzing divisor pairs. For a positive integer ( n ) with prime factorization ( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} ), the total number of positive divisors is given by:", "[\n(d(n) + 1) = (e_1 + 1)(e_2 + 1) \cdots (e_k + 1)\n]", "This formula stems from the fact that each exponent ( e_i ) contributes ( e_i + 1 ) choices (from exponent 0 to ( e_i )) when forming divisors. Since divisors are usually considered in positive integers, there are exactly ( d(n) ) positive divisor pairs ( (a, b) ) such that ( a \cdot b = n ) and ( a > 0, b > 0 ).", "### The Role of Negative Divisor Pairs", "Beyond positive divisors, each positive divisor ( a ) pairs with a corresponding negative divisor ( b = -a ) such that ( (-a)(-b) = ab = n ), provided ( n > 0 ). Thus, for every positive divisor pair ( (a, b) ), there is a negative pair ( (-a, -b) ), yielding another 15 positive-looking but sign-flipped divisor pairs.", "For a number ( n ), the total number of ordered integer divisor pairs—counting both ( (a,b) ) and ( (-a, -b) )—is therefore:", "[\n2 \ imes d(n) = 30 \quad \ ext{(since } d(n) = 15\ ext{)}\n]", "### Example Illustration", "Let’s illustrate with ( n = 36 ). First, factorize:\n[\n36 = 2^2 \ imes 3^2 \quad \Rightarrow \quad d(36) = (2+1)(2+1) = 9\n]", "- Positive pairs: ( (1,36), (2,18), (3,12), (4,9), (6,6) ), and symmetric ( (9,4), (12,3), (18,2), (36,1) ) — but counting unordered pairs, we have 9 total positive divisors, forming 9 positive divisor pairs.\n- Negative pairs: ( (-1,-36), (-2,-18), \ldots, (-36,-1) ), another 9 pairs.", "Total divisor pairs: ( 9 + 9 = 18 ). Here, ( d(36) = 9 ), so total pairs = ( 2 \ imes 9 = 18 ), consistent with the 30 idea in a generalized interpretation (though strictly ( 2 \ imes d(n) ), not always ( 2 \ imes d(n) ) as 30 suggests—this requires clarification).", "Wait: The original claim of 30 total divisor pairs (15 positive, 15 negative) assumes ( d(n) = 15 ). So when ( d(n) = 15 ), total signed divisor pairs are ( 2 \ imes 15 = 30 ), counting all ( (a,b) ) and ( (-a,-b) ) with ( ab = n ), but only if ( n > 0 ) to allow pairwise sign flipping maintaining product ( n ).", "But note: If ( n > 0 ), and ( ab = n ), then ( (-a)(-b) = ab = n ), so yes, each positive pair ( (a,b) ) generates a negative pair ( (-a,-b) ). However, order matters: these are distinct ordered pairs.", "If ( d(n) = 15 ), then:\n- 15 positive ordered pairs ( (a,b) ), ( a > 0, b > 0 )\n- 15 negative ordered pairs ( (-a,-b) ), ( a > 0, b > 0 )", "Total: 30 divisor pairs, all with ( a,b \in \mathbb{Z} ), ( ab = n ).", "This holds only when ( n ) is a positive integer with exactly 15 positive divisors.", "### Numbers with Exactly 15 Positive Divisors", "How can a positive integer ( n ) have exactly 15 positive divisors?\nWe apply the divisor rule:\n[\nd(n) = (e_1 + 1)(e_2 + 1) \cdots (e_k + 1) = 15\n]\nFactor 15: ( 15 = 15 \ imes 1 ) or ( 15 = 5 \ imes 3 )", "Thus, possible forms:\n- ( n = p^{14} ): one prime raised to 14th power\n- ( n = p^4 q^2 ): two distinct primes, exponents 4 and 2", "Examples:\n- ( 2^{14} = 16384 ) → ( d(n) = 15 )\n- ( 2^4 \ imes 3^2 = 16 \ imes 9 = 144 ) → ( d(144) = 5 \ imes 3 = 15 )", "For such ( n ), there are precisely 15 positive divisor pairs ( (a,b) ), and 15 negative divisor pairs ( (-a,-b) ), totaling 30 integer divisor pairs.", "### Summary", "- The number of positive divisors is ( d(n) = (e_1+1)\cdots(e_k+1) = 15 ).\n- Each positive divisor ( a ) forms a negative divisor pair ( (-a,-b) ) for the same product ( ab = n ).\n- Thus, for ( d(n) = 15 ), there are 15 positive and 15 negative ordered divisor pairs.\n- Total: 30 divisor pairs over the integers, reflecting symmetry and the multiplicative structure of divisors.", "Understanding divisor counts not only reveals a number’s internal structure but also explains symmetric behaviors across sign domains—key in number theory, cryptography, and algorithm design.", "---", "TL;DR:\nIf a positive integer has exactly 15 positive divisors, it has 15 unordered positive divisor pairs ( (a,b) ) and 15 symmetric negative pairs ( (-a,-b) ), totaling 30 integer divisor pairs—each satisfying ( ab = n )—demonstrating deep symmetry in divisor function behavior."]