1) Find the remainder of the division – \( \frac{(100!)^{100} - (99!)^{99} + (98!)^{98} - (97!)^{97} + \cdots + (2!)^2 - (1!)}{2! \cdot 3! \cdot 4! \cdot 5!} \)

(a) 19873 (b) 12745 (c) 20731 (d) 9751 (e) None Of These

2) Find the remainder of \( \frac{1223334444\ldots\text{(1000 digits)}}{16} \)

(a) 3 (b) 9 (c) 13 (d) 0 (e) 7

3) Find the remainder of \( \frac{1223334444\ldots\text{(1000 digits)}}{9} \)

(a) 5 (b) 3 (c) 4 (d) 7 (e) 8

4) Find the remainder of the division \( \frac{1121231234\ldots\text{(1000 digits)}}{16} \)

(a) 5 (b) 3 (c) 4 (d) 7 (e) 8

5) Find the remainder of division \( \frac{33 + 3333 + 333333 + \ldots \text{(1000 terms)}}{6} \)

(a) 2 (b) 3 (c) 4 (d) 5 (e) 0

6) Find the remainder of \( \frac{333333333333 + 333333333333333333333333 + \ldots \text{(1000 terms)}}{7} \)

(a) 0 (b) 3 (c) 5 (d) 9 (e) None of these

7) Find the remainder of the division \( \frac{(987654321987654321)^{2000} - (123456789123456789)^{2000}}{8} \)

(a) 7 (b) 3 (c) 0 (d) 2 (e) 1

8) Find the remainder of \( \frac{(1000\ldots\text{(100 zeros)})^{(1000\ldots\text{(100 zeros)})}}{7} \)

(a) 1 (b) 4 (c) 3 (d) 5 (e) 0

9) Find the remainder of division \( \frac{151600}{1309} \)

(a) 1308 (b) 1307 (c) 1 (d) 127 (e) 156

10) Find the remainder when \( N = (7!)^{(6!)^{(5!)^{\cdots^{1!}}}} \) is divided by 76

(a) 518 (b) 529 (c) 228 (d) 329 (e) None of these

11) Find the remainder of \( \frac{1889 \times 1891 \times 1893 \times \ldots \times 1987}{7^8} \)

(a) \({7^8} – 7 (b)\) \({7^8} + 7\) (c) \({7^8} – 1\) (d) \({7^8} + 1\) (e) None of these

12) Let \( N \) be a 9999-digit number in a pattern. If \( R = N \mod 9999 \), find unit digit of \( 7^R \)

(a) 7 (b) 9 (c) 3 (d) 1 (e) None of these

13) \( N = 123456789876543212345678987\ldots \) (1000 digits). Find \( N \mod 16 \)

(a) 14 (b) 12 (c) 8 (d) 5 (e) None of these

14) Find the remainder of \( \frac{173^{(163^{153})}}{17} \)

(a) 7 (b) 8 (c) 9 (d) 10 (e) None of these

15) Find the remainder of \( \frac{(101^{(102^{(103^{\ldots^{109}})}})}{109} \)

(a) 17 (b) 23 (c) 49 (d) 104 (e) None of these

16) Find the remainder of \( \frac{(10^{(100^{(1000^{\ldots^{10^{100}}})})}}{629} \)

(a) 518 (b) 528 (c) 228 (d) 329 (e) None of these

17) Find the remainder of \( \frac{8^{1499}}{343} \)

(a) 177 (b) 279 (c) 204 (d) 211 (e) None of these

18) Find the remainder of \( \frac{(33333\ldots \text{(100 digits)})^{(33333\ldots \text{(100 digits)})}}{65} \)

(a) 41 (b) 18 (c) 23 (d) 36 (e) None of these

19) Find the remainder of \( \frac{149^{149223}}{647} \)

(a) 643 (b) 4 (c) 197 (d) 343 (e) None of these

20) Find the remainder of \( \frac{151^{250270}}{3293} \)

(a) 2068 (b) 1057 (c) 2287 (d) 228 (e) None of these