Finding Remainder of Numbers of the Form a^b

Q. If $5^{99}$ is divided by $13$, then the remainder is

Q. The remainder when $2^{2003}$ is divided by $17$ is

1. $2$
2. $8$
3. $1$
4. $11$

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

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

Q.

Find the remainder when ${\left({32}^{32}\right)}^{32}$ is divided by $7$.

Q.

The number $111.........1(91\mathrm{times})$ is a

1. Even number

2. Prime number

3. Not prime

4. None of these

Q.

The remainder when $7^{103}$ is divided by 25 is:

1. 0

2. 9

3. 18

4. None

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 $x=5^{5^{5^{{.}^{{.}^{{.}^{.}}}}}}\left(24\text{times}5\right)$ is divided by $24$, then the remainder is

Q. The last digit of $(2137{)}^{754}$ is

1. $1$
2. $7$
3. $9$
4. $3$

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

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

Q. The last two digits of the number $(23{)}^{14}$ are

1. $01$
2. $09$
3. $03$
4. $05$

Q. The remainder when $3^{100}$ is divided by $100$ is

1. $01$
2. $21$
3. $41$
4. $61$

Q.

The last three digits in 10! are ______.

1. 500

2. 600

3. 800

4. 700

Q. If the remainder when $x$ is divided by $4$ is $3,$ then the remainder when $(2020+x{)}^{2022}$ is divided by $8$ is

Q. The digit at unit's place in the number $(13{)}^{1225}+(11{)}^{1915}-(23{)}^{1225}$ is equal to

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

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. In the expansion of ${(3^{5x/4}+3^{-x/4})}^n$ the sum of binomial coefficient is $64$. If the term with greatest binomial coefficient exceeds the third term by $(n-1),$ then the number of value(s) of $x$ is

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

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. Let $\alpha >0,\beta >0$ be such that ${\alpha }^3+{\beta }^2=4.$ If the maximum value of the term independent of $x$ in the binomial expansion of ${(a{x}^{\frac{1}{9}}+\beta x^{\frac{1}{6}})}^{10}$ is $10k$, then $k$ is equal to:

1. $176$
2. $336$
3. $352$
4. $84$

Q. The remainder when $2^{120}$ is divided by $7$.

1. $3$
2. $5$
3. $6$
4. $1$

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

1. 2
2. 7
3. 8
4. 0

Q. If the sum of the coefficients of first half terms in the expansion of $(x+y{)}^n$ is $256$, where $n$ is odd positive integer, then the greatest coefficient in the expansion is

Q.

The last two digits of the number $3^{400}$ are $01$. State true or false.

Q.

If 7 divides ${32}^{{32}^{32}}$ then the remainder is ___

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. How many two-digit numbers are divisible by $8$?

1. $12$
2. $10$
3. $11$
4. $13$

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

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

Q. If $8^{83}+6^{83}$ is divided by $49$, then the remainder is

Q. The coefficient of $a^{-6}{b}^4$ in the expansion of ${(\frac{1}{a}-\frac{2b}{3})}^{10}$ is

1. $\frac{120}{27}$
2. $\frac{1120}{27}$
3. $\frac{2010}{27}$
4. $\frac{1128}{27}$

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. $15$