Question:
The remainder obtained when $(\lfloor 1)^2 + (\lfloor 2)^2 + (\lfloor 3)^2 + ... + (\lfloor 100)^2$ is divided by $10^2$ is ;
The remainder obtained when $(\lfloor 1)^2 + (\lfloor 2)^2 + (\lfloor 3)^2 + ... + (\lfloor 100)^2$ is divided by $10^2$ is ;
Updated On: May 14, 2024
- 14
- 17
- 28
- 27
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Terms greater than 5!
i.e., $(5 !)^2, ( 6 !)^2, ..., (100!)^2$ is divisible
by 100
$\therefore$ For terms $ (5 !)^2, ( 6 !)^2,$ ..., $(100!)^2$ remainder is 0.
Now consider $(1 !)^2 + (2 !)^2 + (3 !)^2 + (4 !)^2$
= 1 + 4+ 36+ 576
= 617
When 617 is divided by 100, its remainder is 17.
$\therefore$ Required remainder is 17.
Was this answer helpful?
0
0
Top Questions on Binomial theorem
- The sum of all rational numbers in \( (2 + \sqrt{3})^8 \) is:
- JEE Main - 2025
- Mathematics
- Binomial theorem
Let $ (1 + x + x^2)^{10} = a_0 + a_1 x + a_2 x^2 + ... + a_{20} x^{20} $. If $ (a_1 + a_3 + a_5 + ... + a_{19}) - 11a_2 = 121k $, then k is equal to _______
- JEE Main - 2025
- Mathematics
- Binomial theorem
The term independent of $ x $ in the expansion of $$ \left( \frac{x + 1}{x^{3/2} + 1 - \sqrt{x}} \cdot \frac{x + 1}{x - \sqrt{x}} \right)^{10} $$ for $ x>1 $ is:
- JEE Main - 2025
- Mathematics
- Binomial theorem
- If $ \sum_{r=1}^{9} \left( \frac{r+3}{2^r} \right) \cdot {^9C_r} = \alpha \left( \frac{3}{2} \right)^9 - \beta $, $ \alpha, \beta \in \mathbb{N} $, then $ (\alpha + \beta)^2 $ is equal to
- JEE Main - 2025
- Mathematics
- Binomial theorem
- The sum of all rational terms in the expansion of $ \left( 2 + \sqrt{3} \right)^8 $ is
- JEE Main - 2025
- Mathematics
- Binomial theorem
View More Questions
Questions Asked in KCET exam
- The maximum value of \( z = 3x + 4y \), subject to the constraints \( x + y \leq 40, x + 2y \geq 60 \) and \( x, y \geq 0 \) is:
- KCET - 2025
- Linear Programming Problem
- Given, a current carrying wire of non-uniform cross-section, which of the following is constant throughout the length of the wire?
- KCET - 2025
- Current electricity
- If \( A = \{ x : x \text{ is an integer and } x^2 - 9 \geq 0 \} \), \[ B = \{ x : x \text{ is a natural number and } 2 \leq x \leq 5 \}, \quad C = \{ x : x \text{ is a prime number} \leq 4 \} \] Then \( (B - C) \cup A \) is:
- KCET - 2025
- Set Theory
- The number of four digit even numbers that can be formed using the digits 0, 1, 2 and 3 without repetition is:
- KCET - 2025
- permutations and combinations
- Which of the following methods of expressing concentration are unitless?
- KCET - 2025
- Expressing Concentration of Solutions
View More Questions
Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.