Number properties

1)Find the greatest four digit number which is exactly divisible by 393.

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

2) Which of the following numbers is divisible by 22?

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

3) Find the LCM of 368 and 46.

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Prime factors of 368 = 24 * 23
Prime factors of 46 = 2 * 23
LCM of 368 and 46 = 24 * 23 = 368. Hence(d)

4) LCM and HCF of two numbers are 504 and 8 respectively. If one of the numbers is 56, find the other number.

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LCM * HCF = product of two numbers.
504 * 8 = 56 * x. à x= 72. Hence(c)

5) One-fifth of two-third of four-seventh of a number is 16. What is 35% of that number?

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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)

6) When a number is divided by 43 the remainder is 37. When the same number is divided by 29, what will be the remainder?

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N = 43Q + 37, If Q = 0, N = 37
R(37/29) = 8. Hence(d)

7) When a number is successively divided by 9, 11 and 14, it leaves a remainder 4, 7 and 13 respectively. Find the smallest such number.

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If N is the number and q1, q2, q3 are the successive quotients, then x =9q1 + 4
q1 = 11q2 + 7, q2 = 14q3 + 13. The smallest number will be if q3 = 0, N = 1354. Hence(a).

8) Find the unit digit of 8531327

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The cyclicity of 3 is 4. Remainder when power divided by 4 => 3.
Unit digit of 8531327 = 33 = 27. Hence(c)

9) What is the unit digit of 3984191 + 4376297

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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 3984191 = 4
Unit digit of 4376297 = 6
Unit digit of 3984191 + 4376297 = 4 + 6 = 10. Hence(A)

10) How many factors are there in 548?

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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).

11) Find the product of the factors of 1352.

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Number of factors of 1352 = 23 *132 = (3+1) * (2+1) = 12 factors
Product of the factors = N(no of factors /2) = 1352(12/2) à13526. Hence (c).

12) Find the greatest number of 4 digits which when divided by 7, 9 and 11 leaves 4 as remainder in each case.

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LCM of 7, 9 and 11 = 693
Greatest 4 digit number which is divisible by 693 = 9702
Required number = 9702 + 4 = 9706. Hence(d)

13) Convert binary to octal (101111011)2

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1011110112 = 101 | 111 | 011
= (573)8. Hence(a)

14) Find the remainder of 1438157 when divided by 5.

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Unit digit of 1438157 = 8
R(8/5) = 3. Hence (c).

15) Find the remainder when 7169 divided by 72.

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R(\({71{}{}{}{}{}{}{}{}{}{}{}{} \over {72}}\)) = R(\({(72-1) \over {72}}\))
= R(\({72 \over {72}}\)) - R(\({1 \over {72}}\))
= 0 – 1 + 72 = 71. Hence(b).