Even & Odd Numbers
The classification of integers as even or odd is one of the
simplest but most useful ideas in Number System. Competitive
examinations often test parity indirectly through sums,
products, powers, algebraic expressions and consecutive-number
problems. Understanding the rules allows many questions to be
solved without performing lengthy calculations.
1. What is an Even Number?
An integer that is exactly divisible by 2 is called an
even number.
In other words, an integer n is even if:
n = 2k
where k is an integer.
Examples:
... −10, −8, −6, −4, −2, 0, 2, 4, 6, 8, 10 ...
Important:
Zero is an even number because
0 = 2 × 0.
2. What is an Odd Number?
An integer that is not divisible by 2 is called an
odd number.
Every odd integer can be represented as:
n = 2k + 1
where k is an integer.
Examples:
... −9, −7, −5, −3, −1, 1, 3, 5, 7, 9 ...
The equivalent form 2k − 1 can also represent an
odd integer because k may be any integer.
3. Every Integer is Either Even or Odd
Every integer belongs to exactly one of the two classes:
No integer can be both even and odd at the same time.
Fractions such as 3/2 and decimals such as 2.5 are not classified
as even or odd in the usual Number System definition because
evenness and oddness apply to integers.
4. Identifying Even and Odd Numbers from the Unit Digit
In the decimal number system, an integer is even or odd according
to its last digit.
| Unit Digit |
Nature of Number |
| 0 |
Even |
| 2 |
Even |
| 4 |
Even |
| 6 |
Even |
| 8 |
Even |
| 1 |
Odd |
| 3 |
Odd |
| 5 |
Odd |
| 7 |
Odd |
| 9 |
Odd |
Therefore:
Even unit digits: 0, 2, 4, 6, 8
Odd unit digits: 1, 3, 5, 7, 9
5. Addition Rules for Even and Odd Numbers
The parity of a sum can often be determined without knowing the
actual numbers.
| Operation |
Result |
| Even + Even |
Even |
| Odd + Odd |
Even |
| Even + Odd |
Odd |
| Odd + Even |
Odd |
Examples:
- 8 + 12 = 20 → Even
- 7 + 9 = 16 → Even
- 8 + 7 = 15 → Odd
Shortcut:
A sum is odd when an odd number of odd terms are being added.
A sum is even when an even number of odd terms are being added.
6. Subtraction Rules
The same parity pattern applies to subtraction.
| Operation |
Result |
| Even − Even |
Even |
| Odd − Odd |
Even |
| Even − Odd |
Odd |
| Odd − Even |
Odd |
Examples:
- 18 − 10 = 8 → Even
- 15 − 7 = 8 → Even
- 14 − 5 = 9 → Odd
- 17 − 8 = 9 → Odd
7. Multiplication Rules
The multiplication rules are especially useful in algebraic and
divisibility questions.
| Operation |
Result |
| Even × Even |
Even |
| Even × Odd |
Even |
| Odd × Even |
Even |
| Odd × Odd |
Odd |
Key Rule:
If even one factor of an integer product is even, the entire product
is even.
Therefore, a product can be odd only when
every factor is odd.
8. Why Odd × Odd is Always Odd
Let two odd integers be:
2m + 1 and 2n + 1
Their product is:
(2m + 1)(2n + 1)
= 4mn + 2m + 2n + 1
= 2(2mn + m + n) + 1
This has the form 2k + 1.
Therefore, the product of two odd integers is always
odd.
9. Division Does Not Have a Fixed Even-Odd Rule
Unlike addition, subtraction and multiplication, division does not
always produce an integer. Therefore, parity cannot always be
assigned to the quotient.
Examples:
- 8 ÷ 2 = 4 → Even
- 6 ÷ 2 = 3 → Odd
- 4 ÷ 8 = 1/2 → Neither even nor odd
- 9 ÷ 3 = 3 → Odd
Exam Trap:
Do not assume a universal parity rule for division.
First check whether the quotient is an integer.
10. Powers of Even Numbers
Any positive integral power of an even integer is even.
If n is even:
nk is even for every positive integer k
Examples:
- 23 = 8 → Even
- 62 = 36 → Even
- 105 → Even
11. Powers of Odd Numbers
Any positive integral power of an odd integer remains odd.
Oddpositive integer = Odd
Examples:
- 32 = 9 → Odd
- 53 = 125 → Odd
- 710 → Odd
For positive integral exponents, raising a number to a power does
not change whether its base is even or odd.
12. Square of an Integer and Its Parity
The square of an even integer is even, and the square of an odd
integer is odd.
Therefore:
- If n is even, n2 is even.
- If n is odd, n2 is odd.
This also gives the reverse conclusion:
- If n2 is odd, n must be odd.
- If n2 is even, n must be even.
13. Consecutive Integers Alternate Between Even and Odd
Consecutive integers differ by 1. Therefore their parity alternates.
Example:
8, 9, 10, 11, 12, 13
Their pattern is:
Even, Odd, Even, Odd, Even, Odd
Any two consecutive integers consist of
one even number and one odd number.
14. Product of Consecutive Integers
Since any two consecutive integers contain one even number, their
product is always even.
For any integer n:
n(n + 1) is always even
Examples:
- 4 × 5 = 20
- 7 × 8 = 56
- −3 × (−2) = 6
This property is frequently used in divisibility proofs and
competitive-exam simplification questions.
15. Consecutive Even Numbers
Consecutive even integers differ by 2.
They may be represented as:
2n, 2n + 2, 2n + 4, 2n + 6, ...
Example:
12, 14, 16, 18, 20
If three consecutive even integers are required, we may take:
2n − 2, 2n, 2n + 2
This symmetric form is often convenient when their sum or average
is given.
16. Consecutive Odd Numbers
Consecutive odd integers also differ by 2.
They may be represented as:
2n + 1, 2n + 3, 2n + 5, ...
Three consecutive odd integers may conveniently be represented as:
2n − 1, 2n + 1, 2n + 3
Example:
15, 17, 19
17. Average of Consecutive Even or Odd Numbers
For an odd number of equally spaced consecutive terms, the average
is the middle term.
Example 1:
Average of 12, 14, 16, 18, 20
= 16
Example 2:
Average of 9, 11, 13, 15, 17
= 13
This is a property of equally spaced numbers and is not limited only
to even or odd numbers.
18. Number of Even and Odd Integers in Consecutive Ranges
Since parity alternates among consecutive integers:
-
In any set of an even number of consecutive integers, exactly
half are even and half are odd.
-
In any set of an odd number of consecutive integers, one parity
occurs once more than the other.
Example:
Among 10 consecutive integers, exactly 5 are even and 5 are odd.
Among 11 consecutive integers, either 6 are even and 5 odd, or
5 are even and 6 odd, depending on the starting number.
19. Sum of Two Consecutive Integers
Let two consecutive integers be n and n + 1.
Their sum is:
n + (n + 1)
= 2n + 1
Since 2n + 1 is odd:
The sum of any two consecutive integers is always odd.
20. Sum of Consecutive Even Numbers
The sum of any number of even integers is always even.
Example:
8 + 10 + 12 + 14 = 44
Therefore, regardless of how many even integers are added,
the result remains even.
21. Sum of Odd Numbers — Powerful Parity Shortcut
When odd numbers are added:
- An even number of odd terms gives an even sum.
- An odd number of odd terms gives an odd sum.
Examples:
3 + 5 = 8 → Even
3 + 5 + 7 = 15 → Odd
1 + 3 + 5 + 7 = 16 → Even
Exam Shortcut:
To determine whether a large sum is even or odd, you often need
only count how many odd terms it contains. The actual values may
not need to be added.
22. Parity of n2 and n
An integer and its square always have the same parity.
- If n is even, n2 is even.
- If n is odd, n2 is odd.
Therefore:
n and n2 are either both even or both odd.
Examples:
- 8 is even and 82 = 64 is even.
- 7 is odd and 72 = 49 is odd.
23. n2 + n is Always Even
Consider:
n2 + n
Taking n common:
n2 + n = n(n + 1)
n and n + 1 are consecutive integers. One of them must be even.
Therefore their product is always even.
For every integer n:
n2 + n is always even.
24. n2 − n is Also Always Even
We have:
n2 − n = n(n − 1)
n and n − 1 are consecutive integers, so one of them must be even.
Therefore:
n2 − n is always even for every integer n.
25. n2 + n + 1 is Always Odd
Since n2 + n is always even:
n2 + n + 1
= Even + 1
= Odd
Fast Exam Result:
For every integer n,
n2 + n + 1 is odd.
26. Parity of n2 + 1
The parity of n2 + 1 is opposite to the parity of n.
-
If n is even, n2 is even, so n2 + 1 is odd.
-
If n is odd, n2 is odd, so n2 + 1 is even.
Example:
n = 6 → 62 + 1 = 37 → Odd
n = 5 → 52 + 1 = 26 → Even
27. Cube and Higher Positive Powers Preserve Parity
For any positive integer exponent k:
- If n is even, nk is even.
- If n is odd, nk is odd.
Thus n, n2, n3, n4, ... all have
the same even-odd nature when the exponent is a positive integer.
28. Sum of an Integer and Its Square
Since n and n2 have the same parity:
- Even + Even = Even
- Odd + Odd = Even
Therefore:
n + n2 is always even
29. Difference Between an Integer and Its Square
Similarly:
n2 − n
is always even because n and n2 have the same parity.
This gives a quick parity argument without needing to factor
the expression every time.
30. Square of an Even Number is Divisible by 4
Let an even integer be 2k.
Its square is:
(2k)2 = 4k2
Therefore, the square of every even integer is divisible by 4.
Even n → n2 is divisible by 4.
31. Square of an Odd Number
Let an odd integer be:
2k + 1
Its square is:
(2k + 1)2
= 4k2 + 4k + 1
= 4k(k + 1) + 1
Since k(k + 1) is even, 4k(k + 1) is divisible by 8.
Hence the square of every odd integer leaves remainder 1 when
divided by 8.
Important Result:
For every odd integer n,
n2 = 8m + 1
for some integer m.
Examples:
- 32 = 9 = 8 × 1 + 1
- 52 = 25 = 8 × 3 + 1
- 72 = 49 = 8 × 6 + 1
32. Product of Two Consecutive Integers is Even
For any integer n:
n(n + 1)
is always even because one of two consecutive integers must be even.
This result is frequently hidden inside algebraic expressions.
Example:
If an expression contains x(x + 1), there is no need to know whether
x is even or odd. The product is automatically even.
33. Product of Three Consecutive Integers
Among any three consecutive integers:
- At least one is even.
- Exactly one of the three is divisible by 3.
Therefore:
n(n + 1)(n + 2) is always divisible by 6
This property goes beyond parity, but it is a very useful extension
of the consecutive-number concept.
34. Product of Several Integers — Parity Shortcut
A product of integers is odd only if every factor is odd.
Therefore:
- If at least one factor is even, the product is even.
- If every factor is odd, the product is odd.
Example:
17 × 31 × 42 × 55 × 71
Since 42 is even, the complete product is
even.
In large-product questions, you often need to inspect only whether
an even factor is present.
35. Conditional Parity Rules in Division
Division has no universal even-odd rule, but if it is already known
that the quotient is an integer, some useful conclusions can be made.
Even ÷ Odd
If an even integer divided by an odd integer gives an integer
quotient, that quotient must be even.
Example:
18 ÷ 3 = 6 → Even
Odd ÷ Odd
If an odd integer divided by an odd integer gives an integer quotient,
the quotient must be odd.
Example:
45 ÷ 5 = 9 → Odd
Odd ÷ Even
An odd integer cannot be exactly divisible by an even integer.
If it were, the odd number would equal:
Even × Integer
which would have to be even, producing a contradiction.
Even ÷ Even
Even when the quotient is an integer, it may be even or odd.
Examples:
- 12 ÷ 2 = 6 → Even
- 12 ÷ 4 = 3 → Odd
Remember:
These conclusions assume that the quotient under discussion is an integer.
36. Number of Even Integers from 1 to n
The even integers from 1 to n are:
2, 4, 6, 8, ...
Therefore, the number of even positive integers not exceeding n is:
⌊n/2⌋
where ⌊x⌋ denotes the greatest integer less than or equal to x.
Examples:
- From 1 to 20 → 10 even numbers
- From 1 to 21 → 10 even numbers
37. Number of Odd Integers from 1 to n
The number of odd positive integers not exceeding n is:
⌈n/2⌉
Equivalently:
n − ⌊n/2⌋
Examples:
- From 1 to 20 → 10 odd numbers
- From 1 to 21 → 11 odd numbers
38. Sum of the First n Even Numbers
The first n positive even numbers are:
2, 4, 6, ..., 2n
Their sum is:
2 + 4 + 6 + ... + 2n = n(n + 1)
Since n and n + 1 are consecutive integers, their product is always even.
Therefore, the sum of the first n positive even numbers is always
even.
Example:
First 6 even numbers:
2 + 4 + 6 + 8 + 10 + 12
= 6 × 7
= 42
39. Sum of the First n Odd Numbers
A fundamental identity is:
1 + 3 + 5 + ... + (2n − 1) = n2
Therefore, the sum of the first n positive odd numbers has the same
parity as n.
- If n is even, n2 is even.
- If n is odd, n2 is odd.
Example:
1 + 3 + 5 + 7 + 9 = 25 = 52
40. Parity of the Sum of the First n Natural Numbers
The sum of the first n natural numbers is:
S = n(n + 1)/2
Its parity depends on the value of n modulo 4.
| Form of n |
Parity of n(n + 1)/2 |
| n = 4k |
Even |
| n = 4k + 1 |
Odd |
| n = 4k + 2 |
Odd |
| n = 4k + 3 |
Even |
Shortcut:
1 + 2 + 3 + ... + n is:
Even when n ≡ 0 or 3 (mod 4)
Odd when n ≡ 1 or 2 (mod 4)
41. Factorial and Parity
For a positive integer n:
n! = 1 × 2 × 3 × ... × n
If n ≥ 2, the product contains the factor 2.
Therefore:
n! is even for every integer n ≥ 2.
Also:
1! = 1, which is odd.
42. Sum of an Even and an Odd Expression
Sometimes a lengthy expression can be split into parts whose parity
is already known.
Example:
Find the parity of:
n(n + 1) + 2m + 1
n(n + 1) is always even.
2m + 1 is always odd.
Therefore:
Even + Odd = Odd
This technique can eliminate the need for complete algebraic
expansion.
43. Same-Parity and Different-Parity Numbers
If two integers have the same parity, then:
- Their sum is even.
- Their difference is even.
If two integers have different parity, then:
- Their sum is odd.
- Their difference is odd.
Useful Test:
a and b have the same parity if and only if a − b is even.
44. Parity of a2 − b2
Since a2 has the same parity as a and b2
has the same parity as b:
-
If a and b have the same parity,
a2 − b2 is even.
-
If a and b have different parity,
a2 − b2 is odd.
This can also be seen from:
a2 − b2 = (a − b)(a + b)
45. Can the Sum of Two Integers Determine Their Parity?
If the sum of two integers is odd, then one integer must be even
and the other odd.
If their sum is even, then the two integers have the same parity:
- Both may be even, or
- Both may be odd.
An even sum alone does not prove that both integers are even.
They may both be odd.
46. Can the Product Determine the Parity of the Factors?
If the product of two integers is odd:
Both integers must be odd.
If the product is even:
At least one factor must be even.
However, an even product does not tell us whether the other factor
is even or odd.
Logical Shortcut:
Product odd → every integer factor odd.
Product even → at least one integer factor even.
47. Solved Example: Large Sum
Without calculating the complete sum, determine whether:
17 + 28 + 39 + 46 + 51 + 62 + 73
is even or odd.
Odd terms are:
17, 39, 51, 73
There are four odd terms.
An even number of odd terms gives an even sum.
Therefore, the required sum is even.
48. Solved Example: Algebraic Expression
If n is an integer, determine the parity of:
n2 + 3n + 5
Rewrite:
n2 + 3n + 5
= (n2 + n) + 2n + 5
n2 + n is even.
2n is even.
5 is odd.
Therefore:
Even + Even + Odd = Odd
49. Solved Example: Consecutive Integers
Determine whether the product of 101 and 102 is even or odd.
101 and 102 are consecutive integers.
One of two consecutive integers must be even.
Therefore:
101 × 102 is even.
No multiplication is required.
50. Solved Example: Sum of Odd Integers
Is the sum of 37 odd integers necessarily odd?
Yes.
Since 37 is odd, an odd number of odd terms has an odd sum.
Therefore, the sum is odd.
51. Solved Example: Product Condition
If ab is odd and a and b are integers, what can be concluded?
An integer product can be odd only when every factor is odd.
Therefore:
a and b must both be odd.
52. Solved Example: Sum Condition
If a + b is odd and a and b are integers, what can be concluded?
The sum of integers is odd only when their parities are different.
Therefore:
One of a and b is even and the other is odd.
53. Common Exam Traps
Trap 1:
Zero is an even number, not an odd number.
Trap 2:
Even and odd classification applies to integers.
A non-integer such as 3/2 is neither even nor odd.
Trap 3:
Odd + Odd is even, not odd.
Trap 4:
If even one factor is even, the whole integer product is even.
Trap 5:
Division has no universal parity rule because the quotient may
not even be an integer.
Trap 6:
If a sum is even, the terms need not both be even.
Two odd integers also have an even sum.
Trap 7:
If a product is even, it only proves that at least one factor
is even. It does not prove that every factor is even.
Trap 8:
Consecutive even numbers differ by 2, not by 1.
The same is true for consecutive odd numbers.
Trap 9:
An odd square is not merely odd; it always leaves remainder 1
when divided by 8.
Trap 10:
n(n + 1) is always even regardless of whether n itself is even
or odd.
54. Quick Revision
- An even integer has the form 2k.
- An odd integer has the form 2k + 1.
- Zero is even.
- Every integer is either even or odd, but not both.
- Even unit digits are 0, 2, 4, 6 and 8.
- Odd unit digits are 1, 3, 5, 7 and 9.
- Even + Even = Even.
- Odd + Odd = Even.
- Even + Odd = Odd.
- The same parity rules apply to subtraction.
- Any integer product containing an even factor is even.
- A product is odd only when every integer factor is odd.
- Positive integral powers preserve the parity of their base.
- n and n2 have the same parity.
- n2 + n is always even.
- n2 − n is always even.
- n2 + n + 1 is always odd.
- The square of an even integer is divisible by 4.
- The square of an odd integer leaves remainder 1 when divided by 8.
- Any two consecutive integers contain one even and one odd number.
- n(n + 1) is always even.
- n(n + 1)(n + 2) is always divisible by 6.
- The sum of two consecutive integers is always odd.
- An even number of odd terms has an even sum.
- An odd number of odd terms has an odd sum.
- The sum of the first n even numbers is n(n + 1).
- The sum of the first n odd numbers is n2.
- The number of even positive integers up to n is ⌊n/2⌋.
- The number of odd positive integers up to n is ⌈n/2⌉.
- n! is even for every n ≥ 2.
- If a + b is odd, one of a and b is even and the other odd.
- If ab is odd, both a and b are odd.
- If ab is even, at least one of a and b is even.
Practice these verified previous-year questions based on even and odd
numbers, consecutive numbers, parity, squares and related properties.
Try each question before opening the answer and explanation.
Practice these exam-oriented questions on even and odd numbers, parity,
consecutive numbers, powers and algebraic expressions. Try each question
before opening the answer and explanation.