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
- If the sum, \( \sum_{r=1}^9 \frac{r+3}{2r} \cdot \binom{9}{r} \) is equal to \( \alpha \cdot \left( \frac{3}{2} \right)^9 - \beta \), then the value of \( (\alpha + \beta)^2 \) is equal to:
- JEE Main - 2025
- Mathematics
- Binomial theorem
- The sum of all rational numbers in \( (2 + \sqrt{3})^8 \) is:
- JEE Main - 2025
- Mathematics
- Binomial theorem
- If \( \sum_{r=0}^5 \frac{{}^{11}C_{2r+1}}{2r+2} = \frac{m}{n} \), gcd(m, n) = 1, then \( m - n \) is equal to:
- JEE Main - 2025
- Mathematics
- Binomial theorem
- If \( A \) and \( B \) are the two real values of \( k \) for which the system of equations \( x + 2y + z = 1 \), \( x + 3y + 4z = k \), \( x + 5y + 10z = k^2 \) is consistent, then \( A + B = \): (a) 3
- VITEEE - 2024
- Mathematics
- Binomial theorem
- If the \(2^{\text{nd}}\), \(3^{\text{rd}}\), and \(4^{\text{th}}\) terms in the expansion of \( (x + a)^n \) are 96, 216, and 216 respectively, and \( n \) is a positive integer, then \( a + x \) is:
- AP EAMCET - 2024
- Mathematics
- Binomial theorem
View More Questions
Questions Asked in KCET exam
- The electric current flowing through a given conductor varies with time as shown in the graph below. The number of free electrons which flow through a given cross-section of the conductor in the time interval \( 0 \leq t \leq 20 \, \text{s} \) is
- KCET - 2024
- Current electricity
- A die is thrown 10 times. The probability that an odd number will come up at least once is:
- KCET - 2024
- Probability
- The incorrect statement about the Hall-Heroult process is:
- KCET - 2024
- General Principles and Processes of Isolation of Elements
- Corner points of the feasible region for an LPP are $(0, 2), (3, 0), (6, 0), (6, 8)$ and $(0, 5)$. Let $z = 4x + 6y$ be the objective function. The minimum value of $z$ occurs at:
- KCET - 2024
- Linear Programming Problem
- If a random variable $X$ follows the binomial distribution with parameters $n = 5$, $p$, and $P(X = 2) = 9P(X = 3)$, then $p$ is equal to:
- KCET - 2024
- binomial distribution
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.