Number System Aptitude Questions and Answers:

Answer: (a) \(1\)

Solution: According to polynomial remainder theorem, $$\frac{2^{99}}{8} = \frac{(2^3)^{33}}{8}$$ $$= \frac{8^{33}}{8} = \frac{1}{8} = 1 \ (Remainder)$$

Answer: (a) \(1\)

Solution: According to polynomial remainder theorem, $$\frac{2^{10}}{3} = \frac{2^9 \times 2}{3}$$ $$= \frac{(2^3)^3 \times 2}{3} = \frac{(3 + 5)^3 \times 2}{3}$$ $$= \frac{5^3 \times 2}{3} = \frac{(3 + 2)^3 \times 2}{3}$$ $$= \frac{2^3 \times 2}{3} = 1 \ (Remainder)$$

Answer: (a) \(1\)

Solution: According to polynomial remainder theorem,$$\frac{2^{33}}{7} = \frac{(2^3)^{11}}{7}$$ $$= \frac{(7 + 1)^{11}}{7} = \frac{1}{7} = 1 \ (Remainder)$$

Answer: (b) \(2\)

Solution: According to polynomial remainder theorem,$$\frac{7^{33}}{5} = \frac{(5 + 2)^{33}}{5}$$ $$= \frac{2^{33}}{5} = \frac{(2^3)^{11}}{5} = \frac{(5 + 3)^{11}}{5}$$ $$= \frac{3^{11}}{5} = \frac{3^8 \times 3^3}{5} = \frac{(3^2)^4 \times 3^3}{5}$$ $$= \frac{(5 + 4)^4 \times 3^3}{5} = \frac{4^4 \times 3^3}{5}$$ $$= \frac{(4^2)^2 \times 3^3}{5} = \frac{(5 + 11)^2 \times 3^3}{5}$$ $$= \frac{11^2 \times 3^3}{5} = \frac{(5 + 6)^2 \times 3^3}{5}$$ $$= \frac{6^2 \times 3^3}{5} = \frac{(5 + 1)^2 \times 3^3}{5}$$ $$= \frac{1 \times 3^3}{5} = \frac{27}{5} = 2$$

Answer: (c) \(3\)

Solution: According to polynomial remainder theorem,$$\frac{3^{101}}{8} = \frac{(3^2)^{50} \times 3}{8}$$ $$= \frac{(8 + 1)^{50} \times 3}{8} = \frac{3}{8} = 3 \ (Remainder)$$

Answer: (b) \(2\)

Solution: According to polynomial remainder theorem,$$\frac{8^{101}}{6} = \frac{(6 + 2)^{101}}{6}$$ $$= \frac{2^{101}}{6} = \frac{2^{99} \times 2^2}{6} = \frac{(2^3)^{33} \times 2^2}{6}$$ $$= \frac{(6 + 2)^{33} \times 2^2}{6} = \frac{2^{33} \times 2^2}{6}$$ $$= \frac{(2^3)^{11} \times 2^2}{6} = \frac{(6 + 2)^{11} \times 2^2}{6}$$ $$= \frac{2^{11} \times 2^2}{6} = \frac{2^9 \times 2^2 \times 2^2}{6}$$ $$= \frac{(2^3)^3 \times 2^4}{6} = \frac{(6 + 2)^3 \times 2^3 \times 2}{6}$$ $$= \frac{2^3 \times 2^3 \times 2}{6} = \frac{(6 + 2) \times (6 + 2) \times 2}{6}$$ $$= \frac{2 \times 2 \times 2}{6} = \frac{8}{6} = 2 \ (Remainder)$$

Answer: (d) \(4\)

Solution: According to polynomial remainder theorem,$$\frac{4^{10}}{6} = \frac{2^{20}}{6}$$ $$= \frac{(2^3)^6 \times 2^2}{6} = \frac{(6 + 2)^6 \times 2^2}{6}$$ $$= \frac{2^6 \times 2^2}{6} = \frac{(2^3)^2 \times 2^2}{6}$$ $$= \frac{(6 + 2)^2 \times 2^2}{6} = \frac{4 \times 4}{6}$$ $$= \frac{16}{6} = 4 \ (Remainder)$$

Answer: (d) \(6\)

Solution: According to polynomial remainder theorem,$$\frac{6^5}{30} = \frac{(6^2)^2 \times 6}{30}$$ $$= \frac{(30 + 6)^2 \times 6}{30} = \frac{6^2 \times 6}{30}$$ $$= \frac{(30 + 6) \times 6}{30} = \frac{6 \times 6}{30}$$ $$= \frac{36}{30} = 6 \ (Remainder)$$

Answer: (a) \(23\)

Solution: According to polynomial remainder theorem,$$\frac{7^3}{40} = \frac{7^2 \times 7}{40}$$ $$= \frac{(40 + 9) \times 7}{40}$$ $$= \frac{9 \times 7}{40} = \frac{63}{40} = 23 \ (Remainder)$$

Answer: (d) \(4\)

Solution: According to polynomial remainder theorem,$$\frac{3^{40}}{5} = \frac{(3^2)^{20}}{5}$$ $$= \frac{(5 + 4)^{20}}{5} = \frac{4^{20}}{20}$$ $$= \frac{2^{40}}{5} = \frac{(2^3)^{13} \times 2}{5}$$ $$= \frac{(5 + 3)^{13}}{5} = \frac{3^{13} \times 2}{5}$$ $$= \frac{(3^2)^6 \times 3 \times 2}{5} = \frac{(5 + 4)^6 \times 6}{5}$$ $$= \frac{4^6 \times (5 + 1)}{5} = \frac{4^6}{5}$$ $$= \frac{2^{12}}{5} = \frac{(2^3)^4}{5} = \frac{(5 + 3)^4}{5}$$ $$= \frac{3^4}{5} = \frac{(3^2)^2}{5} = \frac{(5 + 4)^2}{5}$$ $$= \frac{4^2}{5} = \frac{16}{5} = 1 \ (Remainder)$$