The greatest four digit number is 9999. Dividing the number by 393, the remainder is 174. The required number = 9999 – 174 = 9825. Hence(A)

For the number to be divisible by 22, it should be divisible by 2 and 11. Hence(b)

Prime factors of 368 = 2^{4} * 23

Prime factors of 46 = 2 * 23

LCM of 368 and 46 = 2^{4} * 23 = 368. Hence(d)

Prime factors of 46 = 2 * 23

LCM of 368 and 46 = 2

LCM * HCF = product of two numbers.

504 * 8 = 56 * x. à x= 72. Hence(c)

504 * 8 = 56 * x. à x= 72. Hence(c)

Let the number be x.

\({1 \over {5}}({2 \over {3}})\) (\({4x \over {7}}\)) = 16, à x=210\(\)

35% of 210 = 73.5. Hence(b)

\({1 \over {5}}({2 \over {3}})\) (\({4x \over {7}}\)) = 16, à x=210\(\)

35% of 210 = 73.5. Hence(b)

N = 43Q + 37, If Q = 0, N = 37

R(37/29) = 8. Hence(d)

R(37/29) = 8. Hence(d)

If N is the number and q_{1}, q_{2}, q_{3} are the successive quotients, then x =9q_{1} + 4

q_{1} = 11q_{2} + 7, q_{2} = 14q_{3} + 13. The smallest number will be if q_{3} = 0, N = 1354. Hence(a).

q

The cyclicity of 3 is 4. Remainder when power divided by 4 => 3.

Unit digit of 853^{1327} = 3^{3} = 27. Hence(c)

Unit digit of 853

The cyclicity of 4 is 2. If the power is odd number unit digit will be 4 and the power is even number unit digit will be 6.

The cyclicity of 6 is 1

Unit digit of 3984^{191} = 4

Unit digit of 4376^{297} = 6

Unit digit of 3984^{191} + 4376^{297} = 4 + 6 = 10. Hence(A)

The cyclicity of 6 is 1

Unit digit of 3984

Unit digit of 4376

Unit digit of 3984

Prime factors of the number 546 = (2*3*7*13)

Number of factors = (1+1)*(1+1)*(1+1)*(1+1) = 16 factors. Hence(b).

Number of factors = (1+1)*(1+1)*(1+1)*(1+1) = 16 factors. Hence(b).

Number of factors of 1352 = 2^{3} *13^{2} = (3+1) * (2+1) = 12 factors

Product of the factors = N^{(no of factors /2)} = 1352^{(12}^{/2)} à1352^{6}. Hence (c).

Product of the factors = N

LCM of 7, 9 and 11 = 693

Greatest 4 digit number which is divisible by 693 = 9702

Required number = 9702 + 4 = 9706. Hence(d)

Greatest 4 digit number which is divisible by 693 = 9702

Required number = 9702 + 4 = 9706. Hence(d)

101111011_{2} = 101 | 111 | 011

= (573)_{8}. Hence(a)

= (573)

Unit digit of 1438^{157} = 8

R(8/5) = 3. Hence (c).

R(8/5) = 3. Hence (c).

R(\({71{}{}{}{}{}{}{}{}{}{}{}{} \over {72}}\)) = R(\({(72-1) \over {72}}\))

= R(\({72 \over {72}}\)) - R(\({1 \over {72}}\))

= 0 – 1 + 72 = 71. Hence(b).

= R(\({72 \over {72}}\)) - R(\({1 \over {72}}\))

= 0 – 1 + 72 = 71. Hence(b).