Calculating the number of trailing zeros, the last five digits, and the first five digits can all be done separately.

Use a prime factorization property to find the number of trailing zeros. Use modulo to find the last 5 digits. Use a logarithm property to find the first 5 digits.

The number of trailing zeros C is nothing but the number of times the product is completely divisible by 10. Since 2 and 5 are the only prime factors of 10, C will be equal to the minimum number of times 2 or 5 appear in the prime factorization of the product.

Iterate through the integers from left to right. For every integer, keep dividing it by 2 as long as it is divisible by 2 and C occurrences of 2 haven't been removed in total. Repeat this process for 5. Finally, multiply the integer under modulo of 10^5 with the product obtained till now to obtain the last five digits.

The product P can be represented as P=10^(x+y) where x is the integral part and y is the fractional part of x+y. Using the property "if S = A * B, then log(S) = log(A) + log(B)", we can write x+y = log_10(P) = sum(log_10(i)) for each integer i in [left, right]. Once we obtain the sum, the first five digits can be represented as floor(10^(y+4)).