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 Most Important Multiple Choice Questions
 Online Number System Exercise with Correct Answer Key and Solutions
 Useful for all Competitive Exams
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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
Complete remainder
= d₁d₂r₃ + d₁r₂ + r
= 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 twodigit 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
Twothird 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.
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