1) Find the remainder of \( \frac{32^{32^{32}}}{7} \)

(a) 2 (b) 4 (c) 5 (d) 6 (e) None of these

2) Find the remainder of \( \frac{5^{6^{7^{8^{9^{10}}}}}}{8} \)

(a) 5 (b) 1 (c) 3 (d) 6 (e) None of these

3) Find the remainder of \( \frac{101100111000\ldots\text{(100 digits)}}{16} \)

(a) 5 (b) 1 (c) 7 (d) 6 (e) None of these

4) Find the remainder of \( \frac{3^{4^{5^{.\,^{.\,^{.\,^{2001}}}}}}}{7} \)

(a) 3 (b) 2 (c) 6 (d) 4 (e) 5

5) Find the remainder of \( \frac{2^{45000}}{315} \)

(a) 312 (b) 77 (c) 174 (d) 123 (e) 1

6) Find the remainder of \( \frac{(123456789123456789)^7 - 5(123456789123456789)^3 + 4 \cdot 123456789123456789}{120} \)

(a) 112 (b) 111 (c) 113 (d) 114 (e) 0

7) Find the remainder of \( \frac{7^{7^{7^{7^{.\,^{.\,^{.\,^{2001}}}}}}}}{4} \)

(a) 2 (b) 0 (c) 3 (d) 1 (e) None of these

8) Find the remainder of \( \frac{333333\ldots\text{(3333 times)}}{7} \)

(a) 5 (b) 4 (c) 3 (d) 1 (e) 0

9) Find the remainder of \( \frac{81^{82^{83}}}{17} \)

(a) 16 (b) 12 (c) 3 (d) 1 (e) 4

10) Find the remainder of \( \frac{9^9 + 99^{99} + 999^{999} + \cdots + (\text{99...9})^{\text{99...9}}}{6} \)

(a) 2 (b) 4 (c) 1 (d) 0 (e) 3

11) Find the remainder of \( N \equiv 7 \pmod{D},\ N^2 \equiv 1 \pmod{D} \Rightarrow \text{# of valid D} = ? \)

(a) 5 (b) 3 (c) 2 (d) 1 (e) 6

12) Find the remainder of \( \text{How many distinct remainders when } (p_1^2 - p_2^2) \bmod 6,\ p_i > 3\text{ prime} \)

(a) 4 (b) 3 (c) 2 (d) 1 (e) 6

13) Find the remainder of \( \text{How many distinct remainders when } (p_1^3 - p_2^3) \bmod 6,\ p_i > 3\text{ prime} \)

(a) 5 (b) 4 (c) 3 (d) 2 (e) None of these

14) Find the remainder of \( \frac{2n^9 + 3n^6 + 2n^3 - 3n^2 - 4n + 129}{6},\ \text{given } n \equiv 4 \pmod{12} \)

(a) 3 (b) 4 (c) 0 (d) 2 (e) 1

15) Find the remainder of \( \text{If } \frac{18^{181} \cdot N^{200}}{5} \equiv 3,\ \text{find unit digit of } N^2 \)

(a) 1, 4, or 6 (b) 4, 6 or 9 (c) 1, 6 or 9 (d) All above options are correct (e) None of these

16) Find the remainder of \( \frac{3^{3333} + 5^{5555} + 7^{7777}}{13} \)

(a) 3 (b) 5 (c) 7 (d) 9 (e) None of these

17) Find the remainder of \( \frac{3^{33^{333^{3333^{.\,^{.\,^{.\,^{3333}}}}}}}}{455} = R,\ \text{Range for } R? \)

(a) 82 🔽 Show Answer