Step-by-step explanation:

To find the number of values of n for which 10n + 8 is a perfect square, we can use the fact that a perfect square can only have an odd number of factors.

Let's write 10n + 8 as a product of its prime factors:

10n + 8 = 2^3 * (5n + 4)

For 10n + 8 to be a perfect square, (5n + 4) must be a perfect square because 2^3 is already a perfect cube.

Let's write (5n + 4) as a product of its prime factors:

5n + 4 = p1^a1 * p2^a2 * ... * pm^am, where p1, p2, ..., pm are distinct primes.

For (5n + 4) to be a perfect square, all the exponents a1, a2, ..., am must be even.

We can see that 5n + 4 is an odd number because it is the sum of an even number (8) and an odd number (5n). Therefore, we can write:

5n + 4 = p1^(2b1) * p2^(2b2) * ... * pm^(2bm)

where b1, b2, ..., bm are natural numbers.

Substituting this back into the expression for 10n + 8, we get:

10n + 8 = 2^3 * (5n + 4) = 2^3 * p1^(2b1) * p2^(2b2) * ... * pm^(2bm)

To ensure that 10n + 8 is a perfect square, we need all the exponents of the prime factors to be even.

Therefore, we need b1, b2, ..., bm to be odd. This means that each of the prime factors p1, p2, ..., pm must be of the form 5k + 1, where k is a natural number.

In other words, (5n + 4) must be a product of prime factors of the form 5k + 1.

For example, let's take (5n + 4) = (5*1 + 1)(5*2 + 1)(5*3 + 1) = 6 * 11 * 16 = 1056.

Then, we have:

10n + 8 = 2^3 * 6^2 * 11^2 * 16^2 = (2 * 6 * 11 * 16)^2

So, there is at least one solution for n in this case.

To find all possible values of n, we need to consider all possible products of prime factors of the form 5k + 1. This is an infinite set, so there are infinitely many possible values of n that satisfy the condition.

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