An Integer can simply be defined as a number which can be mathematically represented without a fraction. The set of integers is represented by a bold letter ‘Z’. It represents the smallest group of natural numbers, while the algebraic integers can be distinguished from the rational integers is the theory of algebraic numbers. Different programming languages such as Java and C, are used as an integer as a data type.

Some of the different properties of Integers based on multiplication and addition are:

- Distributivity
- Commutativity
- Associativity
- Closure

You can find the RS Aggarwal Class 7 Solutions Chapter 1 Integers is available below:

**1)1. Evaluate: **

**(i) 15 +(- 8) **

15 + (-8) = 7

**(ii) (-16)+9**

(-16)+ 9 = -7

**(iii) (-7) + (-23) **

(-7) + (-23) = -30

**(Iv) (-32) + 47 **

(-32) + 47 = 15

**(v) 53 +(-26) **

53 + (-26) = 27

**(vi) (-48) + (-36)**

(-48) + (-36) = -84

**2) Find the sum of: **

** (i) 153 and -302 **

153 + (-302) = -149

**(ii) 1005 and -277**

1005 + (-277) = 728

**(iii) -2035 and 297 **

(-2035) + 297 = -1738

**(iv) -489 and -324 **

(-489) + (-324) = -813

**(v) -1000 and 438 **

(-1000) + 438 = -562

**(vi) -238 and 500**

(-238) + 500 = 262

**3) Find the additive inverse of:**

**Solution:**

**(i)-83**

Additive inverse of -83 = -(-83) = 83

**(ii)256**

Additive inverse of 256 = -(256) = -256

**(iii)0**

Additive inverse of 0 = -(0) = 0

**(iv)-2001**

Additive inverse of 2001 = -(-2001) = 2001

**4) Subtract:**

**Solution:**

**(i) 28 from -42**

-42 – 28 = (-42) + (-28) = -70

**(ii) -36 from 42**

42 -(-36) = 42 + 36 = 78

**(iii) -37 from -53**

-53 – (-37) = (-53) – (-37) = -16

**(iv) -66 from -34**

-34 – (-66) = -34 + 66 = 32

**(v) 318 from 0**

0 – 318 = -318

**(vi) -153 from -240**

(-240) – (-153) = -87

**(vii) -64 from 0**

0 – (-64) = 0 + 64 = 64

**(viii) -56 from 144**

144 – (-56) = 144 + 56 = 200

**5) Subtract the sum of -1032 and 878 from -34.**

**Solution:**

Sum of -1032 and 878 = -1032 + 878 = -154

Subtracting the sum from -34, we get

-34 – (-154)

= (-34)+ 154

= 120

**6) Subtract -134 from the sum of 38 and -87.**

**Solution:**

First, we will calculate the sum of 38 and -87.

38 + (-87) = -49

Now, subtracting -134 from the sum, we get:

-49 – (-134)

=(-49) + 134

= 85

**7) Fill in the blanks:**

**(i) {(-13) + 27}4- (-41) = (-13) + {27 +(â€¦â€¦..)}**

**(ii) (-26) + {(-49) + (- 83)1 = {(-26)+ (-49)1+ (â€¦..) **

**(iii) 53 + (-37) = (-37) + (â€¦â€¦â€¦.) **

**(iv) (-68) + (-76) = (â€¦â€¦â€¦.) +(- 68)**

**(v) (-72) + (â€¦â€¦..) = -72 **

**(vi) -(-83) =â€¦â€¦â€¦**

**(vii) (-60) – (â€¦â€¦â€¦)= -59 **

**(viii) (-31) +(â€¦â€¦â€¦.) = -40**

**Solution:**

(i) -41 ( Associative property)

(ii) -83 ( Associative property)

(iii) 53 (v Commutative property)

(iv) -76 ( Commutative property)

(v) 0( Additive identity)

(vi) 83 ( Additive inverse)

(vii) (-60) – (-59) = -1

(viii) (-40) – (-31) = -9

**8)Simplify: (-13 – (-27)) + (-25 – (-40)). **(-13 – (-27)) + (-25 – (-40))

= (-13 + 27) + (-25 + 40)

=14 + 15

= 29

**9) Find 36 – (-64) and (-64) -36. Are they equal?**

36-(-64)= 36+64= 100

Now, (-64) – 36 = (-64) + (-36) = -100

Here, 100 is not equal -100

Thus, they are not equal.

**10) Ifa =-8,b=-7,c=6,verify that(a+b)+c=a+(b+c).**

(a+b)+c=(-8+(-7))+6=-15+6=-9

a +(b+c)= – 8 +(-7+6)=-13+(-1)=-9

Hence, (a + b) + c = a + (b + c) [i.e., Property of Associativity]

**11) If a = -9 and b = -6, show that (a – b) Not Equal To (b – a).**

Here, (a – b) = -9 – (-6) = -3

Similarly, (b – a) = -6 – (-9) = 3

:. (a-b) not equal to (b-a)

**12) The sum of two integers is -16. If one of them is 53. find the other.**

Let the other integer be a. Then, we have:

53+a=-16

=>a = -16 – 53 = -69

The other integer is -69.

**13) The sum of two integers is 65. If one of them is -31, find the other.**

Let the other integer be a.

Then, -31 + a= 65

=>a= 65 – (-31) = 96

The other integer is 96.

**14) The difference of an integer a and (-6) is 4. Find the value of a.**

We have:

a – (-6) = 4

=>a=4+(-6)=-2

:. a = -2

**15)Write a pair of integers whose sum gives **

**(i) zero; **

**(ii) a negative integer; **

**(iii) an integer smaller than both the integers; **

**(iv) an integer greater than both the integers; **

**(v) an integer smaller than only one of the integers.**

(i) Consider the integers 8 and -8. Then, we have: 8 + (-8) = 0

(ii) Consider the integers 2 and (-9). Then, we have: 2 + (-9)= -7, which is a negative integer.

(iii) Consider the integers -4 and -5. Then, we have: (-4) + (-5) = -9, which is smaller than -4 and -5.

(iv) Consider the integers 2 and 6. Then, we have: 2 + 6 = 8, which is greater than both 2 and 6.

(v) Consider the integers 7 and -4. Then, we have: 7 + (-4) = 3, which is smaller than 7 only.

**16)For each of the following statements, write (T) for true and (F) for false:**

**(i) The smallest integer is zero. **

**(ii) -10 is greater than -7. **

**(iii) Zero is larger than every negative integer.**

**(iv) The sum of two negative integers is a negative integer. **

**(v) The sum of a negative integer and a positive integer is always a positive integer.**

(i) F (false). -3. -90 and -100 are also integers. We cannot determine the smallest integer. since they are infinite.

(ii) F (false). -10 is less than -7.

(iii) T (true). Al negative integers are less than zero.

(iv) T (true).

(v) F (false). Example: -9 + 2 = -7

**EXERCISE â€“ 1B**

**1)Multiply:
**

**(i) 16 by 9**

**(ii) 18 by -6 **

**(iii) 36 by -11 **

**(iv) -28 by 14 **

**(v) -53 by 18 **

**(vi) -35 by 0 **

**(vii) 0 by -23 **

**(viii) -16 by -12 **

**(ix) -105 by -8 **

**(x) -36 by -50 **

**(xi) -28 by -1 **

**(xii) 25 by -11**

(i) 16 x 9 = 144

(ii) 18 x (-6) = -(18×6) = -108

(iii) 36 x (-11) = – (36×11) = -396

(iv) (-28) x14 = -(28×14) = -392

(v) (-53) X 18 = -(53×18) = -954

(vi) (-35) x 0 = 0

(vii) 0 x (-23) = 0

(viii) (-16) x (-12) = 192

(ix) (-105) x (-8) = 840

(x) (-36) x (-50) = 1800

(xi) (-28) x (-1) = 28

(xii) 25 X (-11) = (25×11) = -275

**2)Find each of the following products: **

**(i) 3x4x(-5) **

**(ii) 2 x (-5)x (-6) (iii) (-5)x (-8)x (-3) **

**(iv) (-6)x 6 x (-10) **

**(v) 7x (-8)x3 **

**(vi) (-7)x (-3)x 4**

(i) 3 x 4 x (-5) = (12) x (-5) = -60

(ii) 2 x(-5)x(-6)=(-10)x(-6)=60

(iii) (-5) x (-8) x (-3) = (-5) x (24) = -120

(iv) (-6)x6 x(-10)=6 x(60)=360

(v) 7 x (-8) x 3 = 21 x (-8) = -168

(vi) (-7) x (-3) x 4 = 21 x 4 = 84

**3)Find each of the following products: **

**(i) (- 4)x (-5)x(- 8)x(-10) **

**(ii) (-6)x (- 5)x (-7) x (-2) x (-3) **

**(iii) (-60)x (-10)x (-5) x (-1) **

**(iv) (-30)x (-20)x (-5) **

**(v) (-3)x (-3)x (-3)x … 6 times **

**(vi) (-5)x (-5) x (-5) x … 5 times **

**(vii) (-1)x (-1)x (-1)x … 200 times **

**(viii) (-1)x (-1)x (-1)x …171 times**

(i) Since the number of negative integers in the product is even. the product will be positive. (4) x (5) x (8) x (10) = 1600

(ii) Since the number of negative integers in the product is odd. the product will be negative. -(6) x (5) x (7) x (2) x (3) = -1260

(iii) Since the number of negative integers in the product is even, the product will be positive. (60) x (10) x (5) x (1) = 3000

(iv) Since the number of negative integers in the product is odd. the product will be negative. -(30) x (20) x (5) = -3000

(v) Since the number of negative integers in the product is even. the product will be positive. (-3)6 = 729

(vi) Since the number of negative integers in the product is odd. the product will be negative. (-5)5 = -3125

(vii) Since the number of negative integers in the product is even the product will be positive. (-1)200= 1

(viii) Since the number of negative integers in the product is odd. the product will be negative. (-1)171 = -1

**4)What will be the sign of the product, if we multiply 90 negative integers and 9 positive integers?**

Multiplying 90 negative integers will yield a positive sign as the number of integers is even.

Multiplying any two or more positive integers always gives a positive integer.

The product of both(the above two cases) the positive and negative integers is also positive.

Therefore, the final product will have a positive sign.

**5) What will be the sign of the product, if we multiply 103 negative integers and 65 positive Integers?**

Multiplying 103 negative integers will yield a negative integer, whereas 65 positive integers will give a positive integer. The product of a negative integer and a positive integer is a negative integer.

**6) Simplify: **

**(i) (-8) x 9 + (- 8) x 7 **

**(ii) 9 x (-13)+ 9 x (-7) **

**(iii) 20 x (-16) + 20 x14 **

**(iv) (-16)x (-15) + (-16) x (-5) **

**(v) (-11)x (-15)+ (-11) x (-25) **

**(vi) 10 x (-12)+5 x (-12) **

**(vii) (-16)x (- 8) + (- 4) x (-8) **

**(viii) (-26) x 72 + (-26) x 28**

(i) (-8) x (9 + 7) [using the distributive law]

= (-8) x 16 = -128

(ii) 9 x (-13 + (-7)) [using the distributive law]

=9 x(-20)=-180

(iii) 20 x (-16 + 14) [using the distributive law]

= 20 x (-2) = -40

(iv) (-16) x (-15 + (-5)) [using the distributive law]

= (-16) x (-20) = 320

(v) (-11) x (-15 +(-25)) [using the distributive law]

= (-11) x (-40) = 440

(vi) (-12) x (10 + 5) [using the distributive law]

= (-12) x 15 = -180

(vii) (-16 + (-4)) x (-8) [using the distributive law]

= (-20) x (-8) = 160

(viii) (-26) x (72 + 28) [using the distributive law]

= (-26) x100 = -2600

**7)Fill In the blanks: **

**(i) (-6)x(……) = 6 **

**(ii) (-18)x (……) = (-18) **

**(iii) (-8) x (-9) = (-9) x (……)**

**(iv) 7 x (-3) = (-3)x (……) **

**(v) I(-5) x x (- 6) = (……)x {3 x (- 6)} **

**(vi) (-5)x(……)=0**

(i) (-6) x (x) = 6

x = 6 / â€“ 6

x = -1

Thus, x = (-1)

(ii) 1 [Multiplicative identity]

(iii) (-8) [Commutative law]

(iv) 7 [Commutative law]

(v) (-5) [Associative law]

(vi) 0 [Property of zero]

**8)In a class test containing 10 questions, 5 marks are awarded for every correct answer and (-2) marks are awarded for every incorrect answer and 0 for each question not attempted. **

**(i) Ravi gets 4 correct and 6 incorrect answers. What is his score? Oil **

**(ii) Reenu gets 5 correct and 5 Incorrect answers. What Is he score? (iii) Heenagets 2 correct and 5 incorrect answers. What is her score?**

**Solution:**

We have 5 marks for correct answer and (-2) marks for an incorrect answer.

Now, we have the following:

(i) Ravi’s score = 4 x 5 + 6 x (-2)

= 20 + (-12) =8

(ii) Reenu’s score = 5 x 5 + 5 x (-2)

= 25 – 10 = 15

(iii) lieena’s score = 2 x 5+ 5 x (-2)

= 10 – 10 = 0

**9)Which of the following statements are true and which are false? **

**(i) The product of a posttive and a negative integer is negative. **

**(ii) The product of two negative integers is a negative integer. **

**(iii) The product of three negative integers is a negative Integer **

**(iv) Every integer when multiplied with -1 gives its multiplicative inverse.**

**(v) Multiplication on integers is commutative. **

**(vi) Multiplication on integers is associative. **

**(vii) Every nonzero integer has a multiplicative inverse as an integer.**

(i) True.

(ii) False. Since the number of negative signs is even, the product will be a positive integer.

(iii) True. The number of negative signs is odd.

(iv) False. a x (-1) = -a, which is not the multiplicative inverse of a.

(v) True. axb=bxa

(vi) True. (axb)xa=ax(bxa)

(vii) False. Every non-zero integer a has a multiplicative inverse 1a, which is not an integer.