Just the explanation
Let’s break down what the explanation means:1.	The Problem Setup:You’re given a string of digits. You need to reduce this string repeatedly by taking every pair
Let’s break down what the explanation means:
	1.	The Problem Setup:
You’re given a string of digits. You need to reduce this string repeatedly by taking every pair of adjacent digits, adding them together, and then taking the result modulo 10 (i.e. keeping only the last digit of the sum). You do this repeatedly until only two digits remain. Finally, you check if these two remaining digits are the same.
	2.	The Key Observation – Connection to Binomial Expansion:
Although it might look like you’re just doing a series of additions, there’s a hidden structure. If you write out how each digit from the original string affects the final two digits, you’ll notice that the process is similar to a binomial expansion. In a binomial expansion (like (a + b)^n), each term has a coefficient (a binomial coefficient) that tells you how many ways that term can be formed.
Here, each digit’s contribution to the final result can be thought of as being multiplied by a certain binomial coefficient—this tells you “how many times” (or with what weight) that digit shows up in the final two digits.
	3.	Working Modulo 10:
Since every operation is done modulo 10, you only care about these coefficients modulo 10. However, 10 is a composite number (10 = 2 × 5), so directly computing binomial coefficients modulo 10 is tricky.
	4.	Efficient Computation with Lucas’ Theorem:
To handle this, you break the problem into two parts:
	•	Compute the binomial coefficients modulo 2.
	•	Compute the binomial coefficients modulo 5.
Lucas’ Theorem is a mathematical tool that makes it easier to compute binomial coefficients modulo a prime (or a prime power). After you get the coefficients modulo 2 and modulo 5, you combine them using the Chinese Remainder Theorem (which is a way to solve congruences) to obtain the coefficients modulo 10.
	5.	Using the Coefficients to Compute the Final Result:
Once you have the relevant binomial coefficients modulo 10, you can use them to determine the contribution of each digit in the original string toward the final two digits. You do this using modular arithmetic, accumulating the contributions into two sums (let’s call them s1 and s2 for the two final digits).
	6.	Checking the Answer:
Finally, after computing these sums (again, modulo 10), you compare the two digits. If they’re the same, you return true; otherwise, false.
	7.	Complexity Considerations:
The explanation also mentions time and space complexity. Although it doesn’t give exact numbers in the snippet, it hints that by using properties of binomial coefficients and Lucas’ Theorem, the solution can be made efficient despite the underlying combinatorial nature of the problem.
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