## Number System problems with solutions

# Number System problems with solutions for goverment exam like Bank,SSC,IBPS

Number System problems with solutions for Bank Exam

**Natural numbers **1, 2, 3, 4, 5, …………………………….. are called natural numbers

**Whole numbers** 0,1, 2, 3, 4, 5, 6, ………………………… are called whole numbers

**Integers** 0, ± 1, ± 2, ± 3, ± 4,………………………… are called integers .

**Rational numbers** Numbers which can be written in the form of p/q, where p and q are integers and q ≠ 0 are called rational numbers.

A number whose decimal expansion is terminating or recurring (recurring means repetition of a set of digits)

**Irrational numbers** Numbers whose decimal expression is non recurring or non terminating are called irrational numbers.

## Number System Question and Solution for Bank Exam like IBPS and SBI PO

**Real numbers** **:** Real number contains all the rational numbers and all the irrational numbers.

**Fraction** A fraction is a rational number written in the form of a/b, where a and b are integers and b ≠ 0.

**Proper fraction** if the denominator of a fraction is greater than the numerator then the fraction is called a proper fraction

**Example :** ¼, ⅔, ⅓

**Improper fraction** if the numerator of a fraction is greater than the denominator then the fraction is called a improper fraction

Example 3/2, 17/5, 9/2

**Divisor, Dividend, Quotient and Remainder**

Suppose we have to divide 17 by 3

Then 3 is the divisor, 17 is the dividend

5 is the quotient and 2 is the remainder

There is a important formula about divisor, dividend, quotient and remainder

**Dividend = divisor quotient****+ remainder**

**Modulus value or absolute value : **The modulus value of a number is equal to that number with positive sign or **(positive value of that number). **The symbol of modulus is x.

**Example : **Ι5Ι = 5, Ι-5.4 Ι = 5.4,Ι Ι-6 Ι= 6

**Some important tricks **

- If a/b < c/d, then a + c/b + d lies between a/b and c/d
- If a and b are two rational numbers such that a < b, then a + b/2 lies between a and b
- The sum of first n natural numbers i.e. 1 +2 + 3 +………+ n = n (n+1)/2

** Example : **1 + 2 + 3 + ………+ 12 = 12 × 13/2 = 78

- The sum of squares of first n natural numbers i.e.12+22+….+n2 = n(n+1)(2n+1)/6

** Example :** 1^{2 }+ 2^{2}** ^{ }**+….+ 15

**= 15 × 16 × 31/6= 1240**

^{2}- The sum of cubes of first n natural numbers i.e. 1
+ 2^{3 }+…….+n^{3 }= n^{3 }(n+1)2/4^{2 }

**Example :** 1** ^{3 }**+ 2

**+…….+12**

^{3 }**= 12**

^{3 }**× 13**

^{2 }**/4= 6084**

^{3}- The sum of first n odd natural numbers = n^2

** Example : **1 + 3 + 5 + ………+ 19 = 10** ^{2 }**= 100

- The sum of first n even natural numbers = n
+ n^{2 }

** Example :** 2 + 4 + 6 + …….. + 40 = 20** ^{2 }**+ 20 = 400 + 20 = 420

- The sum of squares of even natural numbers upto n2= n(n+1)(n+2)/6

** Example :** 2** ^{2 }**+ 4

**+……….+ 10**

^{2 }**= 10 × 11 × 12/6 = 220**

^{2 }- The sum of squares of odd natural numbers up to n^2= n(n+1)(n+2)/6

** Example :** 1** ^{2 }**+ 3

**+………… + 9**

^{2 }**= 9 × 10 × 11/6 = 165**

^{2 }**1** The unit digit of 7** ^{254}** is

- 7
- 9
- 3
- 1

**Answer b) 9**

**Solution** The unit digit of 71= 7

The unit digit of 7** ^{2 }**= 9

The unit digit of 7** ^{3 }**= 3

The unit digit of 7** ^{4}**= 1

The next 4 consecutive power of 7 gives the unit digit 7, 9, 3 and 1 respectively

When we divide 254 by 4, the remainder is 2

And the unit digit of 7** ^{2}**= 9

So, the unit digit of 7^254 is 9

**2** The decimal expression 2.23 is equal to

- 22/9
- 223/99
- 221/99
- 222/99

**Answer c) 221/99**

**Solution** Let x = 2.23

Then, x = 2.232323………… (1)

Multiply by 100 in equation (1)

100x = 223.232323…….. (2)

Subtract (1) from (2)

99x = 221

x = 221/99

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**3** Find the remainder when 2** ^{30 }**+ 3

**+ 4**

^{30 }**+ 5**

^{30 }**divided by 7**

^{30}- 1
- 2
- 3
- 4

**Answer d) 4**

**Solution** when 2** ^{1 }**divided by 7, the remainder is 2

When 2** ^{2}** divided by 7, the remainder is 4

When 2** ^{3}** divided by 7, the remainder is 1

2** ^{30 }**= (2

**)**

^{3}

^{10}When 2** ^{30}** divided by 7 the remainder is same as the remainder when 1^10 divided by 7

2** ^{30}**/7, remainder = 1

Similarly 3** ^{30}**/7, remainder = 1

4** ^{30}**/7, remainder = 1

5** ^{30}**/7, remainder = 1

2** ^{30 }**+ 3

**+ 4**

^{30 }**+5**

^{30 }**/7, remainder is same as the remainder when 1 + 1 + 1 + 1 divided by 7**

^{30}Remainder = 4

**4** When a number is divided by 682, the remainder is 420. What is the remainder when the same number is divided by 31 ?

- 0
- 12
- 17
- 30

**Answer c) 17**

**Solution** since 31 is a factor of 682 i.e. 682 is completely divided by 31.

So, the remainder comes when 420 is divided by 31

Remainder = 17, when 420 divided by 31

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**5 **A rational number lies between 2/7and 3/5 is

- 9/26
- 3/11
- 6/23
- 12/47

**Answer a) ****9/****26**

**Solution** 2 + 3/7 + 5 = 5/12 lies between 2/7 and 3/5

5 + 2/7 + 12 = 7/19 lies between 2/7 and 5/12

7 + 2/19 + 7 = 9/26 lies between 2/7 and 7/19

9/26 lies between 2/7 and 3/5

### Number System Question asked in IBPS exam

**6** The numbers 2272 and 875 are divided by a 3_ digit number N, giving the same remainders. The sum of the digits of N is

- 10
- 11
- 12
- 13
**ssc cgl prelim exam, 04/02/2007**

**Answer a) 10**

**Solution** Let the remainder be x

Then, 2272 – x and 875 – x are completely divide by the N

Then their difference i.e. (2272 – x) – (875 – x) = 1397 is completely divide by the N

1397 = 11×127

Where 11 is prime, which has no other factor than 1 and 11 itself

N= 127

1 + 2 + 7 = 10

**7.** The total number of integers between 100 and 200, which are divisible by both 9 and 6, is

- 5
- 6
- 7
- 8
**ssc cgl prelim exam, 08/02/2004**

**Answer b) 6**

**Solution** If a number is divisible by 9 and 6, then it is divisible by the lcm of 9 and 6 i.e. by 18

108, 126, 144, 162, 180, and 198 are the six numbers between 100 and 200 which are divisible by 18

**8** which of the following is not an irrational number ?

- √214
- π√4
- e/√6
- √8/√2

**Answer d) √****8/√****2**

**Solution √**8/√2 = √4×2/√2

= √4×√2/√2

= √4

= 2

2 is a rational number, hence it is not an irrational number

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