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Simplifying (8 + n)(7 + n)(6 + n)(5 + n)(4 + n)(3 + n)(2 + n) = 0 Multiply (8 + n) * (7 + n) (8(7 + n) + n(7 + n))(6 + n)(5 + n)(4 + n)(3 + n)(2 + n) = 0 ((7 * 8 + n * 8) + n(7 + n))(6 + n)(5 + n)(4 + n)(3 + n)(2 + n) = 0 ((56 + 8n) + n(7 + n))(6 + n)(5 + n)(4 + n)(3 + n)(2 + n) = 0 (56 + 8n + (7 * n + n * n))(6 + n)(5 + n)(4 + n)(3 + n)(2 + n) = 0 (56 + 8n + (7n + n2))(6 + n)(5 + n)(4 + n)(3 + n)(2 + n) = 0 Combine like terms: 8n + 7n = 15n (56 + 15n + n2)(6 + n)(5 + n)(4 + n)(3 + n)(2 + n) = 0 Multiply (56 + 15n + n2) * (6 + n) (56(6 + n) + 15n * (6 + n) + n2(6 + n))(5 + n)(4 + n)(3 + n)(2 + n) = 0 ((6 * 56 + n * 56) + 15n * (6 + n) + n2(6 + n))(5 + n)(4 + n)(3 + n)(2 + n) = 0 ((336 + 56n) + 15n * (6 + n) + n2(6 + n))(5 + n)(4 + n)(3 + n)(2 + n) = 0 (336 + 56n + (6 * 15n + n * 15n) + n2(6 + n))(5 + n)(4 + n)(3 + n)(2 + n) = 0 (336 + 56n + (90n + 15n2) + n2(6 + n))(5 + n)(4 + n)(3 + n)(2 + n) = 0 (336 + 56n + 90n + 15n2 + (6 * n2 + n * n2))(5 + n)(4 + n)(3 + n)(2 + n) = 0 (336 + 56n + 90n + 15n2 + (6n2 + n3))(5 + n)(4 + n)(3 + n)(2 + n) = 0 Combine like terms: 56n + 90n = 146n (336 + 146n + 15n2 + 6n2 + n3)(5 + n)(4 + n)(3 + n)(2 + n) = 0 Combine like terms: 15n2 + 6n2 = 21n2 (336 + 146n + 21n2 + n3)(5 + n)(4 + n)(3 + n)(2 + n) = 0 Multiply (336 + 146n + 21n2 + n3) * (5 + n) (336(5 + n) + 146n * (5 + n) + 21n2 * (5 + n) + n3(5 + n))(4 + n)(3 + n)(2 + n) = 0 ((5 * 336 + n * 336) + 146n * (5 + n) + 21n2 * (5 + n) + n3(5 + n))(4 + n)(3 + n)(2 + n) = 0 ((1680 + 336n) + 146n * (5 + n) + 21n2 * (5 + n) + n3(5 + n))(4 + n)(3 + n)(2 + n) = 0 (1680 + 336n + (5 * 146n + n * 146n) + 21n2 * (5 + n) + n3(5 + n))(4 + n)(3 + n)(2 + n) = 0 (1680 + 336n + (730n + 146n2) + 21n2 * (5 + n) + n3(5 + n))(4 + n)(3 + n)(2 + n) = 0 (1680 + 336n + 730n + 146n2 + (5 * 21n2 + n * 21n2) + n3(5 + n))(4 + n)(3 + n)(2 + n) = 0 (1680 + 336n + 730n + 146n2 + (105n2 + 21n3) + n3(5 + n))(4 + n)(3 + n)(2 + n) = 0 (1680 + 336n + 730n + 146n2 + 105n2 + 21n3 + (5 * n3 + n * n3))(4 + n)(3 + n)(2 + n) = 0 (1680 + 336n + 730n + 146n2 + 105n2 + 21n3 + (5n3 + n4))(4 + n)(3 + n)(2 + n) = 0 Combine like terms: 336n + 730n = 1066n (1680 + 1066n + 146n2 + 105n2 + 21n3 + 5n3 + n4)(4 + n)(3 + n)(2 + n) = 0 Combine like terms: 146n2 + 105n2 = 251n2 (1680 + 1066n + 251n2 + 21n3 + 5n3 + n4)(4 + n)(3 + n)(2 + n) = 0 Combine like terms: 21n3 + 5n3 = 26n3 (1680 + 1066n + 251n2 + 26n3 + n4)(4 + n)(3 + n)(2 + n) = 0 Multiply (1680 + 1066n + 251n2 + 26n3 + n4) * (4 + n) (1680(4 + n) + 1066n * (4 + n) + 251n2 * (4 + n) + 26n3 * (4 + n) + n4(4 + n))(3 + n)(2 + n) = 0 ((4 * 1680 + n * 1680) + 1066n * (4 + n) + 251n2 * (4 + n) + 26n3 * (4 + n) + n4(4 + n))(3 + n)(2 + n) = 0 ((6720 + 1680n) + 1066n * (4 + n) + 251n2 * (4 + n) + 26n3 * (4 + n) + n4(4 + n))(3 + n)(2 + n) = 0 (6720 + 1680n + (4 * 1066n + n * 1066n) + 251n2 * (4 + n) + 26n3 * (4 + n) + n4(4 + n))(3 + n)(2 + n) = 0 (6720 + 1680n + (4264n + 1066n2) + 251n2 * (4 + n) + 26n3 * (4 + n) + n4(4 + n))(3 + n)(2 + n) = 0 (6720 + 1680n + 4264n + 1066n2 + (4 * 251n2 + n * 251n2) + 26n3 * (4 + n) + n4(4 + n))(3 + n)(2 + n) = 0 (6720 + 1680n + 4264n + 1066n2 + (1004n2 + 251n3) + 26n3 * (4 + n) + n4(4 + n))(3 + n)(2 + n) = 0 (6720 + 1680n + 4264n + 1066n2 + 1004n2 + 251n3 + (4 * 26n3 + n * 26n3) + n4(4 + n))(3 + n)(2 + n) = 0 (6720 + 1680n + 4264n + 1066n2 + 1004n2 + 251n3 + (104n3 + 26n4) + n4(4 + n))(3 + n)(2 + n) = 0 (6720 + 1680n + 4264n + 1066n2 + 1004n2 + 251n3 + 104n3 + 26n4 + (4 * n4 + n * n4))(3 + n)(2 + n) = 0 (6720 + 1680n + 4264n + 1066n2 + 1004n2 + 251n3 + 104n3 + 26n4 + (4n4 + n5))(3 + n)(2 + n) = 0 Combine like terms: 1680n + 4264n = 5944n (6720 + 5944n + 1066n2 + 1004n2 + 251n3 + 104n3 + 26n4 + 4n4 + n5)(3 + n)(2 + n) = 0 Combine like terms: 1066n2 + 1004n2 = 2070n2 (6720 + 5944n + 2070n2 + 251n3 + 104n3 + 26n4 + 4n4 + n5)(3 + n)(2 + n) = 0 Combine like terms: 251n3 + 104n3 = 355n3 (6720 + 5944n + 2070n2 + 355n3 + 26n4 + 4n4 + n5)(3 + n)(2 + n) = 0 Combine like terms: 26n4 + 4n4 = 30n4 (6720 + 5944n + 2070n2 + 355n3 + 30n4 + n5)(3 + n)(2 + n) = 0 Multiply (6720 + 5944n + 2070n2 + 355n3 + 30n4 + n5) * (3 + n) (6720(3 + n) + 5944n * (3 + n) + 2070n2 * (3 + n) + 355n3 * (3 + n) + 30n4 * (3 + n) + n5(3 + n))(2 + n) = 0 ((3 * 6720 + n * 6720) + 5944n * (3 + n) + 2070n2 * (3 + n) + 355n3 * (3 + n) + 30n4 * (3 + n) + n5(3 + n))(2 + n) = 0 ((20160 + 6720n) + 5944n * (3 + n) + 2070n2 * (3 + n) + 355n3 * (3 + n) + 30n4 * (3 + n) + n5(3 + n))(2 + n) = 0 (20160 + 6720n + (3 * 5944n + n * 5944n) + 2070n2 * (3 + n) + 355n3 * (3 + n) + 30n4 * (3 + n) + n5(3 + n))(2 + n) = 0 (20160 + 6720n + (17832n + 5944n2) + 2070n2 * (3 + n) + 355n3 * (3 + n) + 30n4 * (3 + n) + n5(3 + n))(2 + n) = 0 (20160 + 6720n + 17832n + 5944n2 + (3 * 2070n2 + n * 2070n2) + 355n3 * (3 + n) + 30n4 * (3 + n) + n5(3 + n))(2 + n) = 0 (20160 + 6720n + 17832n + 5944n2 + (6210n2 + 2070n3) + 355n3 * (3 + n) + 30n4 * (3 + n) + n5(3 + n))(2 + n) = 0 (20160 + 6720n + 17832n + 5944n2 + 6210n2 + 2070n3 + (3 * 355n3 + n * 355n3) + 30n4 * (3 + n) + n5(3 + n))(2 + n) = 0 (20160 + 6720n + 17832n + 5944n2 + 6210n2 + 2070n3 + (1065n3 + 355n4) + 30n4 * (3 + n) + n5(3 + n))(2 + n) = 0 (20160 + 6720n + 17832n + 5944n2 + 6210n2 + 2070n3 + 1065n3 + 355n4 + (3 * 30n4 + n * 30n4) + n5(3 + n))(2 + n) = 0 (20160 + 6720n + 17832n + 5944n2 + 6210n2 + 2070n3 + 1065n3 + 355n4 + (90n4 + 30n5) + n5(3 + n))(2 + n) = 0 (20160 + 6720n + 17832n + 5944n2 + 6210n2 + 2070n3 + 1065n3 + 355n4 + 90n4 + 30n5 + (3 * n5 + n * n5))(2 + n) = 0 (20160 + 6720n + 17832n + 5944n2 + 6210n2 + 2070n3 + 1065n3 + 355n4 + 90n4 + 30n5 + (3n5 + n6))(2 + n) = 0 Combine like terms: 6720n + 17832n = 24552n (20160 + 24552n + 5944n2 + 6210n2 + 2070n3 + 1065n3 + 355n4 + 90n4 + 30n5 + 3n5 + n6)(2 + n) = 0 Combine like terms: 5944n2 + 6210n2 = 12154n2 (20160 + 24552n + 12154n2 + 2070n3 + 1065n3 + 355n4 + 90n4 + 30n5 + 3n5 + n6)(2 + n) = 0 Combine like terms: 2070n3 + 1065n3 = 3135n3 (20160 + 24552n + 12154n2 + 3135n3 + 355n4 + 90n4 + 30n5 + 3n5 + n6)(2 + n) = 0 Combine like terms: 355n4 + 90n4 = 445n4 (20160 + 24552n + 12154n2 + 3135n3 + 445n4 + 30n5 + 3n5 + n6)(2 + n) = 0 Combine like terms: 30n5 + 3n5 = 33n5 (20160 + 24552n + 12154n2 + 3135n3 + 445n4 + 33n5 + n6)(2 + n) = 0 Multiply (20160 + 24552n + 12154n2 + 3135n3 + 445n4 + 33n5 + n6) * (2 + n) (20160(2 + n) + 24552n * (2 + n) + 12154n2 * (2 + n) + 3135n3 * (2 + n) + 445n4 * (2 + n) + 33n5 * (2 + n) + n6(2 + n)) = 0 ((2 * 20160 + n * 20160) + 24552n * (2 + n) + 12154n2 * (2 + n) + 3135n3 * (2 + n) + 445n4 * (2 + n) + 33n5 * (2 + n) + n6(2 + n)) = 0 ((40320 + 20160n) + 24552n * (2 + n) + 12154n2 * (2 + n) + 3135n3 * (2 + n) + 445n4 * (2 + n) + 33n5 * (2 + n) + n6(2 + n)) = 0 (40320 + 20160n + (2 * 24552n + n * 24552n) + 12154n2 * (2 + n) + 3135n3 * (2 + n) + 445n4 * (2 + n) + 33n5 * (2 + n) + n6(2 + n)) = 0 (40320 + 20160n + (49104n + 24552n2) + 12154n2 * (2 + n) + 3135n3 * (2 + n) + 445n4 * (2 + n) + 33n5 * (2 + n) + n6(2 + n)) = 0 (40320 + 20160n + 49104n + 24552n2 + (2 * 12154n2 + n * 12154n2) + 3135n3 * (2 + n) + 445n4 * (2 + n) + 33n5 * (2 + n) + n6(2 + n)) = 0 (40320 + 20160n + 49104n + 24552n2 + (24308n2 + 12154n3) + 3135n3 * (2 + n) + 445n4 * (2 + n) + 33n5 * (2 + n) + n6(2 + n)) = 0 (40320 + 20160n + 49104n + 24552n2 + 24308n2 + 12154n3 + (2 * 3135n3 + n * 3135n3) + 445n4 * (2 + n) + 33n5 * (2 + n) + n6(2 + n)) = 0 (40320 + 20160n + 49104n + 24552n2 + 24308n2 + 12154n3 + (6270n3 + 3135n4) + 445n4 * (2 + n) + 33n5 * (2 + n) + n6(2 + n)) = 0 (40320 + 20160n + 49104n + 24552n2 + 24308n2 + 12154n3 + 6270n3 + 3135n4 + (2 * 445n4 + n * 445n4) + 33n5 * (2 + n) + n6(2 + n)) = 0 (40320 + 20160n + 49104n + 24552n2 + 24308n2 + 12154n3 + 6270n3 + 3135n4 + (890n4 + 445n5) + 33n5 * (2 + n) + n6(2 + n)) = 0 (40320 + 20160n + 49104n + 24552n2 + 24308n2 + 12154n3 + 6270n3 + 3135n4 + 890n4 + 445n5 + (2 * 33n5 + n * 33n5) + n6(2 + n)) = 0 (40320 + 20160n + 49104n + 24552n2 + 24308n2 + 12154n3 + 6270n3 + 3135n4 + 890n4 + 445n5 + (66n5 + 33n6) + n6(2 + n)) = 0 (40320 + 20160n + 49104n + 24552n2 + 24308n2 + 12154n3 + 6270n3 + 3135n4 + 890n4 + 445n5 + 66n5 + 33n6 + (2 * n6 + n * n6)) = 0 (40320 + 20160n + 49104n + 24552n2 + 24308n2 + 12154n3 + 6270n3 + 3135n4 + 890n4 + 445n5 + 66n5 + 33n6 + (2n6 + n7)) = 0 Combine like terms: 20160n + 49104n = 69264n (40320 + 69264n + 24552n2 + 24308n2 + 12154n3 + 6270n3 + 3135n4 + 890n4 + 445n5 + 66n5 + 33n6 + 2n6 + n7) = 0 Combine like terms: 24552n2 + 24308n2 = 48860n2 (40320 + 69264n + 48860n2 + 12154n3 + 6270n3 + 3135n4 + 890n4 + 445n5 + 66n5 + 33n6 + 2n6 + n7) = 0 Combine like terms: 12154n3 + 6270n3 = 18424n3 (40320 + 69264n + 48860n2 + 18424n3 + 3135n4 + 890n4 + 445n5 + 66n5 + 33n6 + 2n6 + n7) = 0 Combine like terms: 3135n4 + 890n4 = 4025n4 (40320 + 69264n + 48860n2 + 18424n3 + 4025n4 + 445n5 + 66n5 + 33n6 + 2n6 + n7) = 0 Combine like terms: 445n5 + 66n5 = 511n5 (40320 + 69264n + 48860n2 + 18424n3 + 4025n4 + 511n5 + 33n6 + 2n6 + n7) = 0 Combine like terms: 33n6 + 2n6 = 35n6 (40320 + 69264n + 48860n2 + 18424n3 + 4025n4 + 511n5 + 35n6 + n7) = 0 Solving 40320 + 69264n + 48860n2 + 18424n3 + 4025n4 + 511n5 + 35n6 + n7 = 0 Solving for variable 'n'. The solution to this equation could not be determined.
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