Number System Questions with Solutions

The Number System is a fundamental area of mathematics that forms the basis for various competitive exams, including the Staff Selection Commission (SSC) examinations. Mastery of Number System Questions is crucial for aspirants aiming to excel in the SSC exams, as it encompasses a range of concepts including divisibility, remainders, HCF and LCM, and number properties.

In this article, we present a series of carefully crafted Number System questions, complete with detailed solutions, specifically designed to align with the SSC exam pattern. These questions not only cover the essential concepts but also challenge your problem-solving skills, providing a comprehensive review of the topic. Whether you’re preparing for the SSC CGL, SSC CHSL, or any other SSC exam, these practice problems will help you strengthen your understanding and improve your performance.

Dive into these problems to enhance your numerical aptitude and build confidence in tackling the Number System questions that you’re likely to encounter in your SSC exams. Here are some Number System questions with solutions tailored for the SSC exam:

1. Arrange the fractions $-\frac{7}{8}, -\frac{2}{9}, \frac{5}{12}, \frac{4}{3}$

​ in ascending order.

Options:

a)

$-\frac{7}{8}, -\frac{2}{9}, \frac{5}{12}, \frac{4}{3}$​
b)

$-\frac{7}{8}, -\frac{2}{9}, \frac{4}{3}, \frac{5}{12}$​
c)

$-\frac{2}{9}, -\frac{7}{8}, \frac{5}{12}, \frac{4}{3}$​
d)

$-\frac{2}{9}, -\frac{7}{8}, \frac{4}{3}, \frac{5}{12}$​

Answer: a)

$-\frac{7}{8}, -\frac{2}{9}, \frac{5}{12}, \frac{4}{3}$​

Detailed Explanation:
To arrange the fractions, first convert them to decimals:

• 
  $-\frac{7}{8} = -0.875$ 

  
  

• 
  $-\frac{2}{9} = -0.222$ 

  
  

• 
  $\frac{5}{12} = 0.417$ 

  
  

• 
  $\frac{4}{3} = 1.333$ 

  
  

Since

$-0.875 < -0.222 < 0.417 < 1.333$, the correct order is

$-\frac{7}{8}, -\frac{2}{9}, \frac{5}{12}, \frac{4}{3}$​.

2. The sum of three fractions is $\frac{11}{24}$

​. Dividing the largest fraction by the smallest gives $\frac{7}{6}$

, which is $\frac{1}{3}$

​ greater than the middle fraction. What is the smallest fraction?

Options:

a)

$\frac{5}{8}$ 

​
b)

$\frac{3}{4}$ 

​
c)

$\frac{5}{6}$ 

​
d)

$\frac{3}{7}$​

Answer: b)

$\frac{3}{4}$​

Detailed Explanation:
Let the fractions be

$x$,

$y$, and

$z$such that

$x < y < z$x<y<z.
Given:

$$z/x = \frac{7}{6}, \quad y = z – \frac{1}{3}$$​Also,

$$x + y + z = \frac{11}{24}$$​Using the given conditions and solving, we find that

$x = \frac{3}{4}$​.

3. Which is the largest fraction among $\frac{2}{3}, \frac{3}{5}, \frac{8}{11}, \frac{11}{17}$

?

Options:

a)

$\frac{8}{11}$ 

​
b)

$\frac{3}{5}$ 

​
c)

$\frac{11}{17}$ 

​
d)

$\frac{2}{3}$​

Answer: a)

$\frac{8}{11}$​

Detailed Explanation:
Convert the fractions to decimals:

• 
  $\frac{2}{3} = 0.66$ 

  32​=0.66

• 
  $\frac{3}{5} = 0.60$ 

  53​=0.60

• 
  $\frac{8}{11} = 0.73$ 

  118​=0.73

• 
  $\frac{11}{17} = 0.65$ 

  1711​=0.65

Thus,

$\frac{8}{11}$​ is the largest fraction.

4. If a number is as much greater than 31 as it is less than 75, what is the number?

Options:

a) 106
b) 44
c) 74
d) 53

Answer: d) 53

Detailed Explanation:
Let the number be

$x$. The condition gives:

$$x – 31 = 75 – x$$
Adding

$x$to both sides:

$$2x = 106$$ 

$$x=53$$So, the number is 53.

5. When 335 is added to 5A7, the result is 8B2. If 8B2 is divisible by 3, what is the largest possible value of A?

Options:

a) 8
b) 2
c) 1
d) 4

Answer: d) 4

Detailed Explanation:
For 8B2 to be divisible by 3, the sum of its digits must be divisible by 3.
Possible values for A are 1, 2, 3, 4, 5, and values for B are 5, 6, 7, 9.
For A = 4 and B = 8, the sum

$8 + 8 + 2 = 18$8+8+2=18 is divisible by 3.
Hence, the largest value of A is 4.

6. The greatest among the numbers $\sqrt[2]{1}, \sqrt[3]{2}, \sqrt[6]{3}$

is:

Options:

a)

$\sqrt[3]{2}$​
b)

$\sqrt[2]{1}$​
c)

$\sqrt[6]{3}$​
d)

$\sqrt[2]{1}$​

Answer: d)

$\sqrt[3]{2}$​

Detailed Explanation:
To compare these numbers, we can express each in a common base. Converting all to the same base:

• 
  $\sqrt[2]{1} = 1$ 

  
  

• 
  $\sqrt[3]{2} = 1.26$ 

  
  

• 
  $\sqrt[6]{3} = 1.20$ 

  
  

Thus,

$\sqrt[3]{2}$​ is the greatest.

7. The least number of five digits which has 123 as a factor is:

Options: a) 10037
b) 10086
c) 10081
d) 10063

Answer: b) 10086

Detailed Explanation:
The smallest five-digit number is 10000. Dividing 10000 by 123 leaves a remainder of 37. Therefore, the number must be

$10000 + (123 – 37) = 10086$ 

8. The greatest value among the fractions $\frac{2}{7}, \frac{1}{3}, \frac{5}{6}, \frac{3}{4}$

is:

Options:

a)

$\frac{3}{4}$ 

​
b)

$\frac{5}{6}$ 

​
c)

$\frac{1}{3}$ 

​
d)

$\frac{2}{7}$​

Answer: b)

$\frac{5}{6}$​

Detailed Explanation:
Convert the fractions to decimals:

• 
  $\frac{2}{7} = 0.286$ 

  
  

• 
  $\frac{1}{3} = 0.33$ 

  
  

• 
  $\frac{5}{6} = 0.833$ 

  
  

• 
  $\frac{3}{4} = 0.75$ 

  
  

Thus,

$\frac{5}{6}$ is the greatest fraction.

9. The difference between the greatest and the least prime numbers which are less than 100 is:

Options:

a) 96
b) 97
c) 94
d) 95

Answer: d) 95

Detailed Explanation:
The greatest prime number less than 100 is 97, and the least is 2. The difference is

$97 – 2 = 95$ 

10. What is the arithmetic mean of the first ‘n’ natural numbers?

Options:

a)

$\frac{n(n + 1)}{2}$ 

​
b)

$\frac{n + 1}{2}$ 

​
c)

$\frac{n(2n + 1)}{2}$2

​
d)

$2(n+1)$ 

Answer: b)

$\frac{n + 1}{2}$​

Detailed Explanation:
The arithmetic mean of the first

$n$natural numbers is calculated using the formula:

$$\text{Arithmetic Mean} = \frac{\text{Sum of the first } n \text{ numbers}}{n} = \frac{\frac{n(n+1)}{2}}{n} = \frac{n+1}{2}$$​

Number System Questions

11. The least number that should be added to 2055, so that the sum is exactly divisible by 27 is:

Options:

a) 28
b) 24
c) 27
d) 31

Answer: b) 24

Detailed Explanation:
To find the remainder when dividing 2055 by 27:

• 
  $2055 \div 27$ 

  gives a remainder of 3. So, the number that should be added is $27 – 3 = 24$ 

  .

12. The difference between the greatest and the least four-digit numbers that begin with 3 and end with 5 is:

Options:

a) 999
b) 900
c) 990
d) 909

Answer: c) 990

Detailed Explanation:
The greatest four-digit number starting with 3 and ending with 5 is 3995, and the smallest is 3005. The difference is

$3995 – 3005 = 990$.

13. When a number is divided by 361, it gives a remainder of 47. If the same number is divided by 19, what will be the remainder?

Options:

a) 3
b) 8
c) 9
d) 1

Answer: c) 9

Detailed Explanation:
Since 361 is divisible by 19, the remainder when dividing 47 by 19 is the same.

$47 \div 19$gives a remainder of 9.

14. In a farm, there are cows and hens. If the heads are counted, they are 180, and if the legs are counted, they are 420. How many cows are there?

Options:

a) 130
b) 150
c) 50
d) 30

Answer: d) 30

Detailed Explanation:
Let

$x$x be the number of cows and

$180 – x$ be the number of hens. Since cows have 4 legs and hens have 2 legs:

$$4x + 2(180 – x) = 420$$ 

$$4x + 360 – 2x = 420$$ 

$$2x=60\Rightarrow x=30$$So, there are 30 cows.

15. Weight of a bucket when filled fully with water is 17 kg. If the weight of the bucket when partially filled with water is 13.5 kg, what is the weight of the empty bucket?

Options:

a) 12 kg
b) 8 kg
c) 10 kg
d) 7 kg

Answer: c) 10 kg

Detailed Explanation:
Let

$x$be the weight of the empty bucket and

$y$y the weight of the water. Then:

$$x + y = 17 \quad \text{(full bucket)}$$ 

$$x+0.5y=13.5\text{(partially filled)}$$ 

Subtracting the second equation from the first:

$$0.5y = 3.5 \quad \Rightarrow \quad y = 7 \text{ kg}$$
So,

$x = 17 – 7 = 10$kg.

16. The smallest number of five digits exactly divisible by 476 is:

Options:

a) 47600
b) 10000
c) 10476
d) 10472

Answer: d) 10472

Detailed Explanation:
The smallest five-digit number is 10000. When divided by 476, the remainder is 4. Therefore, the required number is:

$$10000 + (476 – 4) = 10000 + 472 = 10472$$ 

17. Which of the following is the smallest fraction? $\frac{6}{7}, \frac{5}{6}, \frac{7}{8}, \frac{4}{5}$

​

Options:

a)

$\frac{6}{7}$ 

​
b)

$\frac{4}{5}$ 

​
c)

$\frac{5}{6}$ 

​
d)

$\frac{7}{8}$​

Answer: b)

$\frac{4}{5}$​

Detailed Explanation:
Convert the fractions to decimals:

• 
  $\frac{6}{7} = 0.857$ 

  
  

• 
  $\frac{5}{6} = 0.833$ 

  
  

• 
  $\frac{7}{8} = 0.875$ 

  
  

• 
  $\frac{4}{5} = 0.8$ 

  
  

Thus,

$\frac{4}{5}$ is the smallest fraction.

18. Which of the following is the smallest fraction? $\frac{8}{25}, \frac{7}{23}, \frac{11}{23}, \frac{14}{53}$

​

Options:

a)

$\frac{8}{25}$ 

​
b)

$\frac{7}{23}$ 

​
c)

$\frac{11}{23}$ 

​
d)

$\frac{14}{53}$​

Answer: d)

$\frac{14}{53}$ 

Detailed Explanation:
Convert the fractions to decimals:

• 
  $\frac{8}{25} = 0.32$ 

  
  

• 
  $\frac{7}{23} = 0.30$ 

  
  

• 
  $\frac{11}{23} = 0.478$ 

  
  

• 
  $\frac{14}{53} = 0.264$ 

  
  

Thus,

$\frac{14}{53}$ is the smallest fraction.

19. Which of the following is the smallest fraction? $\frac{8}{15}, \frac{14}{33}, \frac{7}{13}, \frac{11}{13}$

1​

Options:

a)

$\frac{8}{15}$ 

​
b)

$\frac{14}{33}$ 

​
c)

$\frac{7}{13}$ 

​
d)

$\frac{11}{13}$​

Answer: b)

$\frac{14}{33}$​

Detailed Explanation:
Convert the fractions to decimals:

• 
  $\frac{8}{15} = 0.533$ 

  
  

• 
  $\frac{14}{33} = 0.424$ 

  
  

• 
  $\frac{7}{13} = 0.538$ 

  
  

• 
  $\frac{11}{13} = 0.846$ 

Thus,

$\frac{14}{33}$is the smallest fraction.

20. The smallest possible three-place decimal number is:

Options:

a) 0.012
b) 0.123
c) 0.111
d) None of the above

Answer: a) 0.012

Detailed Explanation:
A three-place decimal number has three digits after the decimal point. The smallest possible value is 0.001, so among the given options, 0.012 is the smallest.

21. Which of the following is the smallest fraction? $\frac{9}{13}, \frac{17}{26}, \frac{28}{29}, \frac{33}{52}$

​

Options:

a)

$\frac{33}{52}$ 

​
b)

$\frac{17}{26}$ 

​
c)

$\frac{9}{13}$ 

​
d)

$\frac{28}{29}$​

Answer: a)

$\frac{33}{52}$​

Detailed Explanation:
Convert the fractions to decimals:

• 
  $\frac{9}{13} = 0.692$ 

  
  

• 
  $\frac{17}{26} = 0.654$ 

  
  

• 
  $\frac{28}{29} = 0.966$ 

  
  

• 
  $\frac{33}{52} = 0.635$ 

  
  

Thus,

$\frac{33}{52}$ is the smallest fraction.

22. Which of the following is the smallest fraction? $\frac{7}{6}, \frac{7}{9}, \frac{4}{5}, \frac{5}{7}$

​

Options:

a)

$\frac{7}{6}$ 

​
b)

$\frac{7}{9}$ 

​
c)

$\frac{4}{5}$ 

​
d)

$\frac{5}{7}$ 

​

Answer: d)

$\frac{5}{7}$​

Detailed Explanation:
Convert the fractions to decimals:

• 
  $\frac{7}{6} = 1.167$ 

  
  

• 
  $\frac{7}{9} = 0.778$ 

  
  

• 
  $\frac{4}{5} = 0.8$ 

  
  

• 
  $\frac{5}{7} = 0.714$ 

  
  

Thus,

$\frac{5}{7}$ is the smallest fraction.

23. The difference between the greatest and the least four-digit numbers that begin with 3 and end with 5 is:

Options: a) 999
b) 900
c) 990
d) 909

Answer: c) 990

Detailed Explanation:
The greatest four-digit number starting with 3 and ending with 5 is 3995, and the smallest is 3005. The difference is

$3995 – 3005 = 990$ 

24. A number when divided by 2736 leaves the remainder 75. If the same number is divided by 24, then the remainder is:

Options:

a) 12
b) 3
c) 0
d) 23

Answer: b) 3

Detailed Explanation:
Since 2736 is divisible by 24, the remainder when 75 is divided by 24 is the same.

$75 \div 24$gives a remainder of 3.

25. If the sum of the digits of any integer lying between 100 and 1000 is subtracted from the number, then the result is always:

Options:

a) Divisible by 6
b) Divisible by 2
c) Divisible by 9
d) Divisible by 5

Answer: c) Divisible by 9

Detailed Explanation:
Let the number be

$100a + 10b + c$100a+10b+c where

$a$a,

$b$b, and

$c$c are digits. The sum of digits is

$a + b + c$a+b+c.
Subtracting the sum from the number:

$$100a+10b+c-(a+b+c)=99a+9b$$This expression is always divisible by 9.