**Problem Analysis**

Let's represent the number of beads Joyce has as "x". We are given two conditions:

1. When Joyce makes groups of 9 beads, 8 beads are left out.
2. When Joyce makes groups of 8 beads, 6 beads are left out.

We can translate these conditions into equations:

1. x ≡ 8 (mod 9)
2. x ≡ 6 (mod 8)

Now, we need to find the value of "x" that satisfies both equations.

**Solving the Equations**

To solve these congruence equations, we can use the Chinese Remainder Theorem (CRT). CRT states that if we have a system of congruence equations:

x ≡ a (mod m)
x ≡ b (mod n)

where m and n are coprime (i.e., they have no common factors), then the solution is given by:

x ≡ (a * n * inv(n, m) + b * m * inv(m, n)) (mod m * n)

where inv(a, b) represents the modular multiplicative inverse of a modulo b.

In our case, m = 9 and n = 8. We need to find the modular inverses inv(9, 8) and inv(8, 9).

**Finding Modular Inverses**

To find the modular inverse inv(9, 8), we can use the Extended Euclidean Algorithm. The algorithm states that for any two integers a and b, there exist integers x and y such that:

a * x + b * y = gcd(a, b)

In our case, a = 9 and b = 8. We can apply the Extended Euclidean Algorithm to find x and y.

9 * x + 8 * y = gcd(9, 8)

By applying the algorithm, we find that x = -1 and y = 1. Since we are only interested in the modular inverse, we take the modulo of x and y by their respective moduli:

x ≡ -1 (mod 8)
y ≡ 1 (mod 9)

Therefore, inv(9, 8) = -1 (mod 8).

Similarly, applying the Extended Euclidean Algorithm to find inv(8, 9), we find inv(8, 9) = 8 (mod 9).

**Applying the Chinese Remainder Theorem**

Now that we have found the modular inverses, we can apply the Chinese Remainder Theorem formula to find the solution for "x":

x ≡ (8 * 8 * inv(8, 9) + 6 * 9 * inv(9, 8)) (mod 9 * 8)

Simplifying the expression, we get:

x ≡ (64 - 54) (mod 72)
x ≡ 10 (mod 72)

Therefore, the solution for "x" is x ≡ 10 (mod 72).

**Finding the Range of Possible Values for "x"**

Since we know that Joyce has between 25 and 75 colored beads, we can find the range of possible values for "x" within this range.

x ≡ 10 (mod 72)

We can write this congruence equation as:

x = 10 + 72k

where k is an integer.

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