Integers

Integers

Integers are a collection of numerical values including positive numbers, zero and negative numbers.

Examples of integers include −100, −55, −20, −4, 0, 5, 10, 100, 1000, 10000, etc.

These integers can be visually represented on a number line, as shown below:

Properties of Integers

a. Closure Property: The closure property holds for addition, subtraction and multiplication with integers. In other words, when you perform these operations on two integers, the result is also an integer.

Examples: a) −5 + 4 = −1 (Integer)
b) 5 − 4 = 1 (Integer)
c) −5 × 4 = −20 (Integer)

b. Commutative Property: The commutative property applies to addition and multiplication in the set of integers. It means that the order of the integers does not affect the result.

For any two integers a and b,

a + b = b + a

Example: 4 + (−7) = (−7) + 4

Hence , 4 + (−7) = −3 and (−7) + 4 = −3.

a × b = b × a

Example: 4 × (−7) = (−7) × 4 

Hence , 4 × (−7) = −28  and (−7) × 4 = −28.

c. Associative Property: The associative property holds for addition and multiplication in the set of integers. It means that the grouping of integers does not affect the result.
For any three integers a, b and c,

a + (b + c) = (a + b) + c

Example: 3 + (2 + (−4)) = (3 + 2) + (−4)

Hence, LHS = 3 + (2 + (−4))

= 3 + (2 − 4)
= 3 + (−2)
= 3 − 2
= 1

RHS = (3 + 2) + (−4) 

= (5) + (−4) 
= 5 −4 
= 1

a × (b × c) = (a × b) × c

Example: 3 × (2 × (−4)) = (3 × 2) × (−4)

Hence, LHS = 3 × (2 × (−4)) = 3 × (−8) = −24

RHS = (3 × 2) × (−4) = (6) × (−4) = −24

d. Distributive Property: The distributive property applies to multiplication over addition and subtraction for integers. It allows you to distribute a factor across the sum or difference of integers.

For any three integers a, b and c,

a × (b + c) = a × b + a × c

Example: −4 × (3 + 2) = (−4 × 3) + (−4 × 2)

Hence, LHS = −4 × (3 + 2) = −4 × (5) = −20

RHS = (−4 × 3) + (−4 × 2) ) 

= (−12) + (−8) 
= −12 − 8 
= −20

a × (b − c) = a × b − a × c

Example: −4 × (3 − 2) = (−4 × 3) − (−4 × 2)

Hence, LHS = −4 × (3 − 2) = −4 × (1) = −4

RHS = (−4 × 3) − (−4 × 2) 

= (−12) − (−8)
= −12 + 8
= −4

e. Identity Element: “0” is called the additive identity and “1” is called the multiplicative identity. In integers, when you add 0 to any integer or multiply any integer by 1, it remains unchanged.

For any integer a,

a + 0 = 0 + a = a

Examples: 7 + 0 = 0 + 7 = 7

a × 1 = 1 × a = a

Examples: 7 × 1 = 7 × 1 = 7

f. Multiplication by Zero: For any integer a, multiplying it by 0 results in 0 and the order of multiplication does not matter.

For any integer a, a × 0 = 0 × a = 0.

Example: 7 × 0 = 0 × 7 = 0

Rules for Divisibility

1. Divisibility by 2: A number is divisible by 2 if its last digit is 0, 2, 4, 6 or 8.
2. Divisibility by 3: To check if a number is divisible by 3, add up its digits. If the sum is divisible by 3, the number itself is divisible by 3.
3. Divisibility by 6: If a number is divisible by both 2 and 3, it is also divisible by 6.
4. Divisibility by 4: Examine the last two digits of a number. If these two digits form a number that is divisible by 4, then the entire number is divisible by 4.
5. Divisibility by 8: Examine the last three digits of a number. If these three digits create a number that is divisible by 8, then the whole number is divisible by 8.
6. Divisibility by 5: A number is divisible by 5 if its unit digit is either 0 or 5.
7. Divisibility by 7: To determine divisibility by 7, repeatedly double the unit digit and subtract it from the original number. Continue this process until you have a single-digit result. If that result is 0 or 7, then the original number is divisible by 7.
8. Divisibility by 9: Add up the digits of a number. If the sum is divisible by 9, then the number itself is divisible by 9.
9. Divisibility by 10: A number is divisible by 10 if its unit digit is 0.
10. Divisibility by 11: Calculate the sum of the digits at even places and the sum of the digits at odd places. Find the difference between these two sums. If the result is 0 or 11, then the number is divisible by 11.

Special Divisibility Principles

1. If a number is divisible by another number, it is also divisible by each of the factors of the second number.
2. If a number is divisible by two co-prime numbers, it is also divisible by the product of those two numbers.
3. If two given numbers are individually divisible by a specific number, then their sum is also divisible by that same number.
4. If two given numbers are individually divisible by a particular number, then their difference is also divisible by that same number.

Example 1: Solve the given expression:
      21 + (−13) × 17 − (−121)       

a) −59
b) −79
c) 59
d) 79

Answer: b) −79

Explanation: 21 + (−13) × 17 − (−121)
= 21 + (−221) − (−121)
= 21 − 221 + 121
= 142 − 221
= −79 

Example 2: In an olympiad exam, 4 marks are given for every correct answer and 1 mark is deducted for every incorrect answer. Xavier scored 27 marks though he got 11 correct answers. How many incorrect answers had he attempted?

a) 11
b) 13
c) 15
d) 17

Answer: d) 17

Explanation: Marks awarded for every correct answer = +4
Mark deducted for every incorrect answer = −1
Xavier scored 27 marks but he got 11 correct answers.
Marks scored by Xavier for correct answers = 4 × 11 = 44
But he scored 27 marks.
Marks deducted for incorrect answers = 44 − 27 = 17
Number of incorrect answers attempted = 17 ÷ 1 = 17