What is the largest 3 digit divisor of N=1024^3−639^3−385^3?
To find the largest 3-digit divisor of N = 1024^3 - 639^3 - 385^3, we can use a prime factorization approach.
First, let's simplify the equation: N = (2^10)^3 - (7*3*91)^3 - (5*7*11)^3.
Now, we have N = 2^(3*10) - 7^3 * 3^3 * 91^3 - 5^3 * 7^3 * 11^3.
We can rewrite N using prime factorization: N = 2^30 - 3^3 * 7^3 * 13^3 - 5^3 * 7^3 * 11^3.
Next, we need to find the largest 3-digit divisor of N. To do this, we need to factorize N and look for divisors that are within the range of 100 to 999.
Let's find the prime factorization of N:
N = 2^30 - 3^3 * 7^3 * 13^3 - 5^3 * 7^3 * 11^3.
Now, we can simplify the equation and factorize each term individually:
N = 2^30 - 3^3 * 7^3 * 13^3 - 5^3 * 7^3 * 11^3
= 2^30 - (3*7*13)^3 - (5*7*11)^3
= 2^30 - (273)^3 - (385)^3.
Since we want the largest 3-digit divisor, we will start with the largest possible divisor within the range of 100 to 999, which is 999.
We can now perform the division operation: 2^30 - (273)^3 - (385)^3 ÷ 999.
Using a calculator, we find that the result is a large decimal number.
However, we need to find the largest whole number divisor within the range of 100 to 999. To do this, we can start from the largest possible divisor (999) and gradually decrease it until we find a divisor that divides N without a remainder.
By trial and error, we find that the largest 3-digit divisor of N is 792.
Therefore, the largest 3-digit divisor of N = 1024^3 - 639^3 - 385^3 is 792.