Finding Remainder of Numbers of the Form a^b

Q.

If $(2021{)}^{3762}$ is divided by $17$, then the remainder is

Q.

The number of ways of dividing $52$ cards amongst four players equally, are?

1. $\frac{52!}{{\left(13!\right)}^4}$

2. $\frac{52!}{\left[{\left(13!\right)}^2.4!\right]}$

3. $\frac{52!}{\left[{\left(12!\right)}^2.4!\right]}$

4. None of these

Q. If $6^n-5n,n\in N$ is divided by $25$, then the remainder is

1. $1$
2. $3$
3. $5$
4. $0$

Q.

The sum of $i-2-3i+4+...$upto $100$ terms, where$i=\sqrt{-1}$ is

1. $50\left(1-i\right)$

2. $25\left(1+i\right)$

3. $100\left(1-i\right)$

4. $25i$

Q.

The number of ways of distributing $52$ cards among four players so that three players have $17$ cards each and the fourth player has just one card?

1. $\frac{52!}{3!{\left(17!\right)}^3}$

2. $52!$

3. $\frac{52!}{17!}$

4. $\frac{52!}{{\left(17!\right)}^2}$

Q. For any integer $n>1,$ which of the following is/are correct?

1. The unit place digit of $\sum _{r=0}^{100}r!$ is $4$
2. The unit place digit of $2^{2^n}$ is $6$
3. The unit place digit of $\sum _{r=0}^{100}r!+2^{2^n}$ is $0$
4. The unit place digit of $\sum _{r=0}^{100}r!+2^{2^n}$ is $8$

Q. If ${1690}^{2608}+{2608}^{1690}$ is divided by $7$, then the remainder is

1. $1$
2. $5$
3. $0$
4. $2$

Q.

Evaluate the expression $5\sqrt[3]{729}–24$ .

Q. If ${1690}^{2608}+{2608}^{1690}$ is divided by $7$, then the remainder is

1. $5$
2. $2$
3. $0$
4. $1$

Q.

If $7^{103}$ is divided by 25 then the remainder is ___

Q. If $6^n-5n,n\in N$ is divided by $25$, then the remainder is

1. $1$
2. $3$
3. $5$
4. $0$

Q. If $S={99}^n+1,n>1,n\in N$, then which of the following is/are correct?

1. When $n$ is odd, then the last two digits is $00$
2. When $n$ is odd, then the last two digits is $02$
3. When $n$ is even, then the last two digits is $02$
4. When $n$ is even, then the last two digits is $22$

Q. The remainder left out when $8^{2n}-(62{)}^{2n+1}$is divided by 9 is :

1. 2
2. 7
3. 8
4. \N

Q. If $1+2+2^2+2^3+\dots +2^{1999}$ is divided by $5$, then the remainder is

1. $3$
2. $5$
3. $0$
4. $1$

Q. If the fractional part of the number $\frac{2^{403}}{15}$ is $\frac{k}{15}$, then $k$ is equal to :

1. $4$
2. $8$
3. $14$
4. $6$

Q. If {x} represents the fractional part of x, then $\left\{\frac{5^{200}}{8}\right\}$ is

1. $\frac{1}{4}$
2. $\frac{1}{8}$
3. $\frac{3}{8}$
4. $\frac{5}{8}$

Q. If $n\in N,f\left(n\right)={37}^{n+2}+{16}^{n+1}+{30}^n$, then

1. $f\left(n\right)+1$ is divisible by $3$
2. $f\left(n\right)$ is divisible by $3$
3. $f\left(n\right)$ is divisible by $7$
4. $f\left(4\right)$ is divisible by $5$

Q. For any integer $n>1,$ which of the following is/are correct?

1. The unit place digit of $\sum _{r=0}^{100}r!$ is $4$
2. The unit place digit of $2^{2^n}$ is $6$
3. The unit place digit of $\sum _{r=0}^{100}r!+2^{2^n}$ is $0$
4. The unit place digit of $\sum _{r=0}^{100}r!+2^{2^n}$ is $8$

Q.

Which is greater?

${\left(7\right)}^{3}or{\left(3\right)}^7$

Q. If $2^{4n+4}-15n-16,n\in N$ is divided by $225$, then the remainder is

1. $0$
2. $224$
3. $1$
4. $15$