1^2+2^2+3^2+.......+10^10

when divide by 5 whats the remainder i made no typo that's the question i proff read

To find the remainder when the expression 1^2 + 2^2 + 3^2 + ... + 10^10 is divided by 5, you could calculate the value of the expression and then find the remainder. However, since the expression involves a large number of terms and large exponents, it is impractical to perform the calculations manually.

Fortunately, there is a mathematical principle called the Remainder Theorem that can help us simplify the calculation. According to the Remainder Theorem, when an integer is divided by another integer, the remainder is the same as the remainder of the sum of the remainders of its individual terms.

In this case, we can observe a pattern in the remainders of the individual terms:
- When 1 is divided by 5, the remainder is 1. (1 mod 5 = 1)
- When 2 is divided by 5, the remainder is 4. (2 mod 5 = 2)
- When 3 is divided by 5, the remainder is 4. (3 mod 5 = 3)
- And so on...

Since the remainder repeats every 5 terms (1, 4, 4, 3, 2), we need to find the sum of these remainders for a total of 10 terms. The sum is calculated as follows:
1 + 4 + 4 + 3 + 2 + 1 + 4 + 4 + 3 + 2 = 28.

Now we divide this sum by 5 to find the remainder:
28 mod 5 = 3.

Therefore, the remainder when the expression is divided by 5 is 3.