Prove that A= 3410^n +3000^n -2964^n -1439^n is a multiple of 2007 for all positive integer values of n.

We observe that:

A = 3410^n +3000^n -2964^n -1439^n

= (3410^n -2964^n) - (3000^n-1439^n)

Now, (3410^n -2964^n) is divisible by 3410 -2964 = 446;

while, (3000^n-1439^n) is divisible by 3000-1439 =1561

Accordingly, A is divisible by GCD(446, 1561) = 223

Again, A = 3410^n +3000^n -2964^n -1439^n

= (3410^n -1439^n) + (3000^n-2964^n)

Now, (3410^n -1439^n) is divisible by 3410 -1439 = 2961;

while, (3000^n - 2964^n) is divisible by 3000-2469 =531

Accordingly, A is divisible by GCD(531, 2961) = 9

Consequently, A is divisible by LCM(223,9) = 223*9 = 2007, whenever A is a positive integer.