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- Question 1 of 29
##### 1. Question

Two different numbers when divided by the same divisor, left remainder 11 and 21 respectively, and when their sum was divided by the same divisor, remainder was 41. What is the divisor?

##### Hint

Divisor = [Sum of remainders]

– [ Remainder when sum is divided]

= 11 + 21 – 4 = 28

- Question 2 of 29
##### 2. Question

A number when divided by a divisor, left remainder 23. When twice of the number was divided by the same divisor, remainder was 11. Find the divisor.

##### Hint

Let number be N.

Then, N = Divisor × Q₁ + 23

2N = Divisor × Q₂ + 11, where Q₁ and Q₂ are quotients respectively.

- Question 3 of 29
##### 3. Question

A number when divided by 5 leaves a remainder 3. What is the remainder when the square of the same number is divided by 5?

##### Hint

Let the number be 5q + 3, where q is quotient

Now (5q + 3)² = 25q² + 30q + 9

= 25q² + 30q + 5 + 4

= 5[5q² + 6q + 1] + 4

Hence, remainder is 4

- Question 4 of 29
##### 4. Question

A number when successively divided by 7 and 8 leaves the remainders 3 and 5 respectively. What is the remainder when the same number is divided by 56?

##### Hint

56 = d₁ × d₂

∴ required remainder = d₁r₂ + r₁

where d₁ = 7 and r₁ = 3 and r₂ = 5

- Question 5 of 29
##### 5. Question

A number being successively divided by 3, 5 and 8 leaves 1,2 and 4 as remainders respectively. What are the remainders if the order of divisors be reversed?

##### Hint

Let the quotient be q when divided by 8.

n= 3{5(8q+4)+2}+1

=3(40q+22)+1

=120q+66+1

=120q+67Now, if it is divided by 8.

We have n=120q+67=8(15q+8)+3, remainder 3.

And if 15q+8 is divided by 5, we get remainder 3.

And if 3q+1 is divided by 3, we get remainder 1.

Hence, this all gives us the remainders as 3, 3 and 1.Trick: Complete remainder

= d₁d₂r₃ + d₁r₂ + r₁

= 3 × 5 × 4 + 3 × 2 + 1 = 67

Divided 67 by 8, 5 and 3, the remainders are 3, 3, 1. - Question 6 of 29
##### 6. Question

A boy multiplied a certain number x by 13. He found that the resulting product consisted of all nines entirely. Find the smallest value of x.

##### Hint

By actual division, we find that 999999 is exactly divisible by 13. The quotient 76923 is the required number.

- Question 7 of 29
##### 7. Question

A boy had to divide 49471 by 210. He made a mistake in copying the divisor and obtained his quotient as 246 with a remainder 25. What divisor did the boy copy?

##### Hint

By division Algorithm,

49471 = 246 × D + 25

⇒ D = 201

- Question 8 of 29
##### 8. Question

A certain number is divided by 385 by division by factors. The quotient is 102, the first remainder is 4, the second is 6 and the third is 10. Find the number.

##### Hint

Let the number be z. Now

385 = 5 × 7 ×11

x = 11 × 102 + 10 = 1132

y = 7x + 6 = 7 × 1132 + 6 = 7930

z = 5y + 4 = 5 × 7930 + 4 = 39654

- Question 9 of 29
##### 9. Question

Which digits should come in place of * and $ if the number 62684*$ is divisible by both 8 and 5?

##### Hint

Since the given number is divisible by 5, so 0 or 5 must come in place of $. But, a number ending with 5 is never divisible by 8. So, 0 will replace $.

Now, the number formed by the last three digits is 4*0, which becomes divisible by 8, if * is replaced by 4.

Hence, digits in place of * and $ are 4 and 0 respectively.

- Question 10 of 29
##### 10. Question

The smallest number that must be added to 803642 in order to obtain a multiple of 11 is:

##### Hint

On dividing 803642 by 11, we get remainder = 4.

∴ Required number to be added = (11 – 4) = 7.

- Question 11 of 29
##### 11. Question

A number was divided successively in order by 4, 5 and 6. The remainders were respectively 2, 3 and 4. The number is

##### Hint

z = 6 × 1 + 4 = 10

y = 5 × 10 + 3 = 53

x = 4 × 53 + 2 = 214

- Question 12 of 29
##### 12. Question

The least number which must be subtracted from 6709 to make it exactly divisible by 9 is:

##### Hint

On dividing 6709 by 9, we get remainder = 4.

∴ Required number to be subtracted = 4.

- Question 13 of 29
##### 13. Question

What least number must be subtracted from 427398 so that the remaining number is divisible by 15?

##### Hint

On dividing 427398 by 15, we get remainder = 3.

∴ Required number to be subtracted = 3.

- Question 14 of 29
##### 14. Question

When a number is divided by 31, the remainder is 29. When the same number is divided by 16, what will be the remainder?

##### Hint

Number = (31 × Q) + 29.

Given data is inadequate.

- Question 15 of 29
##### 15. Question

A number A4571203B is divisible by 18. Find the value of A and B.

##### Hint

The number is divisible by 18 i.e., it has to be divisible by 2 and 9.

∴ B may be 0, 2, 4, 6, 8.

A + 4 + 5 + 7 + 1 + 2 + 0 + 3 + B

= A + B + 22.

A + B could be 5, 14 (as the sum can’t exceed 18, since A and B are each less than 10).

So, A and B can take the values of 6, 8.

- Question 16 of 29
##### 16. Question

What is the sum of all two-digit numbers that give a remainder of 3 when they are divided by 7?

##### Hint

Number is of the form

= 7n + 3; n = 1 to 13

So,

- Question 17 of 29
##### 17. Question

If a, a + 2 and a + 4 are prime numbers, then the number of possible solutions for a is

##### Hint

a, a + 2, a + 4 are prime numbers.

Put value of ‘a’ starting from 3, we will have 3, 5 and 7 as the only set of prime numbers satisfying the given relationships.

- Question 18 of 29
##### 18. Question

A boy multiplies 987 by a certain number and obtains 559981 as his answer. If in the answer, both 9’s are wrong but the other digits are correct, then the correct answer will be:

##### Hint

987 = 3 × 7 ×47

So, required number must be divisible by each one of 3, 7, 47.

None of the numbers in (a) and (b) are divisible by 3, while (d) is not divisible by 7.

∴ Correct answer is (c).

- Question 19 of 29
##### 19. Question

There is one number which is formed by writing one digit 6 times (e.g. 111111, 444444 etc.). Such a number is always divisible by:

##### Hint

Since 111111 is divisible by each one of 7, 11 and 13, so each one of given type of numbers is divisible by each one of 7, 11, and 13. as we may write, 222222 = 2 × 111111, 333333 = 3 × 111111, etc.

- Question 20 of 29
##### 20. Question

The remainder when 7⁸⁴ is divided by 342 is :

##### Hint

= (7³)²⁸/(7³ – 1)

= {(7³)²⁸ – 1 + 1}/(7³ – 1)

= {(7³)²⁸ – 1}/(7³ – 1) + 1/(7³ – 1)

((7³)²⁸ – 1) / (7³ – 1) is always divisible as it is in the form of (x

^{n}– y^{n}) / (x – y), hence the remainder is 1. - Question 21 of 29
##### 21. Question

How many numbers are there between 300 and 400 in which 7 occurs only once?

##### Hint

The required numbers are 307, 317, 327, 337, 347, 357, 367, 370, 371, 372, 373, 374, 375, 376, 378, 379, 387, 397.

Hence there are 18 numbers.

- Question 22 of 29
##### 22. Question

The four integers next lower than 81, and the four next higher than 81, are written down and added together, this sum is divisible by,

##### Hint

Here, number of integers next higher and next lower are same (=4).

Now, since 81 is divisible by 9, therefore, the sum is divisible by 9

- Question 23 of 29
##### 23. Question

Two-third of a number is thirty less than the original number. The number is,

##### Hint

Let the original number is x. Then

- Question 24 of 29
##### 24. Question

How many numbers, lying between 1 and 500, are divisible by 13?

##### Hint

∴

- Question 25 of 29
##### 25. Question

If 5432*7 is divisible by 9, then the digit in place of * is

##### Hint

A number is divisible by 9 if the sum of its digits is divisible by 9.

Here 5 + 4 + 3 + 2 + * + 7 = 21 + *

So, the digit in place of * is 6

- Question 26 of 29
##### 26. Question

If the fractions ½, ⅔, ⁵⁄₉, ⁶⁄₁₃ and ⁷⁄₉ are arranged in ascending order of their values, which one will be the fourth?

##### Hint

Decimal equivalents of given fractions:

½ = 0.5; ⅔ = 0.67; ⁵⁄₉ = 0.56; ⁶⁄₁₃ = 0.46; ⁷⁄₉ = 0.78

∴ 0.46 < 0.5 < 0.56 < 0.67 < 0.78

⁶⁄₁₃ < ½ < ⁵⁄₉ < ⅔ < ⁷⁄₉

∴ Fourth fraction = ⅔

- Question 27 of 29
##### 27. Question

If the following fractions ⅞, ⅘, ⁸⁄₁₄, ⅗ and ⅚ are arranged in descending order which will be the last in the series?

##### Hint

Decimal equivalents of fractions

⅞ = 0.875, ⅘= 0.8, ⁸⁄₁₄ = 0.57, ⅗= 0.6, ⅚ = 0.83

∴ 0.875 > 0.83 > 0.8 > 0.6 > 0.57

∴ ⅞ > ⅚ > ⅘ > ⅗ > ⁸⁄₁₄

- Question 28 of 29
##### 28. Question

If the fractions ⅖, ¾, ⅘, ⁵⁄₇ and ⁶⁄₁₁ are arranged in ascending order of their values, which one will be the fourth?

##### Hint

Decimal equivalent of given fractions:

⅖ = 0.4; ¾ = 0.75; ⅘ = 0.8; ⁵⁄₇ = 0.714; ⁶⁄₁₁ = 0.545

Clearly, 0.4 < 0.545 < 0.714 < 0.75 < 0.8

∴ ⅖ < ⁶⁄₁₁ < ⁵⁄₇ < ¾ < ⅘

- Question 29 of 29
##### 29. Question

If all the fractions ⅗, ⅛, ⁸⁄₁₁, ⁴⁄₉, ²⁄₇, ⁵⁄₇ and ⁵⁄₁₂ are arranged in the descending order of their values, which one will be the third?

##### Hint

⁸⁄₁₁ = 0.727, ⁵⁄₇ = 0.714, ⅗ = 0.6, ⁴⁄₉ = 0.44, ⁵⁄₁₂ = 0.416, ²⁄₇ = 0.285, ⅛ = 0.125

Descending order :

⁸⁄₁₁, ⁵⁄₇, ⅗, ⁴⁄₉, ⁵⁄₁₂, ²⁄₇, ⅛

So, ⅗ is the third.