Number N = 26 * 55 * 76 * 107; how many factors of N are even numbers?

Prime factorization: N = 2^6 * 5^5 * 7^6 * 10^7 = 2^(6+7) * 5^(5+7) * 7^6 = 2^13 * 5^12 * 7^6

Total number of factors: (13 + 1) * (12 + 1) * (6 + 1) = 14 * 13 * 7 = 1274

Number of odd factors (exclude the factor 2): (12 + 1) * (6 + 1) = 13 * 7 = 91

Number of even factors: total factors − odd factors = 1274 − 91 = 1183

Answer: 1183