Explain in English linear Chinese remainder theorem this problem below

The Chinese Remainder Theorem (CRT) provides a way to solve systems of simultaneous congruences with different moduli. It states that if you have a set of congruences:

x ≡ a₁ (mod m₁)
x ≡ a₂ (mod m₂)
...
x ≡ aₖ (mod mₖ)

where m₁, m₂, ..., mₖ are pairwise coprime (i.e., the greatest common divisor of any two is 1), then there exists a unique solution x modulo M, where M = m₁ × m₂ × ... × mₖ.

To solve a problem using the CRT, follow these steps:

1. Calculate M: Find the product of all moduli, M = m₁ × m₂ × ... × mₖ.
2. Find Mᵢ: For each i, calculate Mᵢ = M / mᵢ.
3. Find the modular inverse: For each i, find the inverse of Mᵢ modulo mᵢ, denoted as yᵢ, such that Mᵢ × yᵢ ≡ 1 (mod mᵢ).
4. Construct the solution: The solution is given by:

x ≡ Σ (aᵢ × Mᵢ × yᵢ) (mod M)

5. Simplify: Reduce x modulo M to find the smallest non-negative solution.

This theorem is particularly useful in number theory and cryptography. If you have a specific problem to apply this to, please share it!