Explanation:

The prime factorization of 2⁶ * 5⁵ * 7⁶ * 10⁷ is 2¹³ * 5¹² * 7⁶.
The total number of factors of N = 14 * 13 * 7
We need to find the total number of even factors.
For this, let us find the total number of odd factors and then subtract this from the total number of factors.
Any odd factor will have to be a combination of powers of only 5 and 7.
Total number of odd factors of 2¹³ * 5¹² * 7⁶ = (12 + 1) * (6 + 1) = 13 * 7
Total number of factors = (13 + 1) * (12 + 1) * (6 + 1)
Total number of even factors = 14 * 13 * 7 - 13 * 7
Number of even factors = 13 * 13 * 7 = 1183