solve this question and explain answer---**Question Stem:** For each nonnegative integer n we define A_n = 2^(3n) + 3^(6n+2) + 5^(6n+2). Find the greatest common divisor of the numbers A_0, A_1, ..., A_{1999}.

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For each nonnegative integer n, we define A_n equals 2 to the power of 3n, plus 3 to the power of 6n plus 2, plus 5 to the power of 6n plus 2. Our goal is to find the greatest common divisor of all the numbers A_0, A_1, A_2, up to A_1999. When finding the GCD of many numbers, we typically look for patterns in how these numbers behave when divided by potential common factors. Let's calculate the first few values of A_n to identify patterns. For A_0, we have 2 to the 0 plus 3 squared plus 5 squared, which equals 1 plus 9 plus 25, giving us 35. For A_1, we get 2 cubed plus 3 to the 8th plus 5 to the 8th, which equals 8 plus 6561 plus 390625, totaling 397194. We can see that both values are divisible by certain numbers, which will help us find the greatest common divisor. Let's rewrite A_n in a more manageable form by factoring out common terms. Starting with A_n equals 2 to the 3n plus 3 to the 6n plus 2 plus 5 to the 6n plus 2, we can factor this as 2 to the 3n plus 3 squared times 3 to the 6n plus 5 squared times 5 to the 6n. This becomes 2 to the 3n plus 9 times 3 to the 6n plus 25 times 5 to the 6n. We can further rewrite this as 2 to the 3n plus 9 times 3 to the 6th power to the n plus 25 times 5 to the 6th power to the n. Since 3 to the 6th equals 729 and 5 to the 6th equals 15625, our final form is A_n equals 2 to the 3n plus 9 times 729 to the n plus 25 times 15625 to the n. This form makes modular arithmetic analysis much easier. To find the greatest common divisor of A_0, A_1, up to A_1999, we need to find the largest number d such that d divides all A_n. This is equivalent to finding d such that A_n is congruent to 0 modulo d for all n from 0 to 1999. Our general approach will be to examine A_n modulo small primes p. If d divides all A_n, then A_n is congruent to A_0 modulo d for all n. We'll use our factored form to make these modular calculations more efficient and find constraints on the possible values of d. Now let's perform detailed modular arithmetic to find constraints on the GCD. Using our factored form A_n equals 2 to the 3n plus 9 times 729 to the n plus 25 times 15625 to the n, we'll analyze this modulo 5 and modulo 7. For modulo 5: 2 cubed is congruent to 3, 729 is congruent to 4, and 15625 is congruent to 0 modulo 5. This gives us A_n congruent to 0 modulo 5 for all n. For modulo 7: 2 cubed is congruent to 1, 729 is congruent to 1, and 15625 is congruent to 4 modulo 7. This shows that A_n is congruent to 3 plus 4 times 4 to the n modulo 7, which equals 0 for all n. Since both 5 and 7 divide all A_n, we know that 35 divides all A_n.