Remainder Theorem

Q. What is the remainder when ${}_{20}{51}^{97}$ is divided by 17?

1. 15
2. 10
3. 7
4. 9

Q. When x is divided by 6, remainder obtained is 3. Find the remainder when $x^4+x^3+x^2+x+1$ is divided by 6.

1. 4
2. 1
3. 3
4. 5

Q. A certain number 'C' when divided by $N_1$ it leaves a remainder of 13 and when it is divided by $N_2$ it leaves a remainder of 1, where $N_1$ and $N_2$ are the positive integers. Then the value of $N_1+N_{2}is,if\frac{N_1}{N_2}=\frac{5}{4}$:

1. 27
2. 54
3. 36
4. can't be determined uniquely

Q. The remainder when n is divided by 3 is 1 and the remainder when (n + 1) is divided by 2 is 1. The remainder when (n - 1) is divided by 6 is :

1. 2
2. 3
3. 5
4. none of (a), (b), (c)

Q. If x is a natural number and 4 < x < 50, then the largest n, such that n! would always divide: $x(x^2-1)(x^2-4)(x^2-9)(x+4)$ is___

Q. What is the remainder when 77, 777... up to 56 digits is divided by 19?

1. 1
2. 13
3. 7
4. 9

Q. Find the remainder when (17) (9!) + 2 (18!) is divided by (9!) 17408___

Q. When a number 'N' is divided by a proper divisor D' then it leaves a remainder of 14 and if the thrice of that number i.e., 3N is divided by the same divisor D, the remainder comes out to be 8. Again if the 4 times of the same number i.e., '4N' is divided by D the remainder will be :

1. 35
2. 22
3. can't be determined
4. 5

Q. Let N be a positive integer not equal to 1. Then, none of the numbers 2, 3, ..., N is a divisor of (N! - 1). Thus, we can conclude that

1. none of the foregoing statement is necessarily correct.
2. (N! - 1) is a prime number.
3. at least, one of the numbers (N + 1), (N + 2)... (N! - 2) is a divisor of (N! - 1).
4. the smallest number between N and N!, which is a divisor of (N! + 1), is a prime number.

Q. What is the remainder when ${}_{55}{56}^{57}$ is divided by 17?

1. 1
2. 4
3. 13
4. 17