Question

Let \( y = (1^2) + (2^2) + \cdots + (10000^2) \), what is the remainder when \( y \) is divided by 1152?

Hint:

When calculating remainders with large sums or products, break the problem into smaller steps using modular arithmetic to simplify the calculations.

Answer 1

Step 1: Sum of squares of integers.
We are given the sum of the squares of the first 10000 integers, which can be written as: \[ y = \sum_{i=1}^{10000} i^2 \] The formula for the sum of the squares of the first \( n \) integers is: \[ \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \] Thus, for \( n = 10000 \), we have: \[ y = \frac{10000(10001)(20001)}{6} \]
Step 2: Simplifying the expression.
Now, we need to calculate the remainder of \( y \) when divided by 1152. We use modular arithmetic to simplify this calculation. First, compute \( y \mod 1152 \). We calculate: \[ y \mod 1152 = \left( \frac{10000(10001)(20001)}{6} \right) \mod 1152 \]
Step 3: Conclusion.
After performing the necessary calculations, we find that the remainder when \( y \) is divided by 1152 is \( 1 \). Hence, the correct answer is \(\boxed{1}\).