Odd Numbers

Odd numbers or integers are part of whole numbers that are partially divisible into pairs. Thus all numbers except the multiples of 2 are odd numbers. They are in the form of 2k+1, where k ∈ Z (integers) are called odd numbers.

Some examples are 1, 3, 5, 7, and so on. They are just the opposite of even numbers. Odd numbers can be represented on a number line:

Odd Numbers on a Number Line

Looking at the examples, we can consider whether all odd numbers are prime.

No, it is not. However, all prime numbers are odd except 2.

List of Odd Numbers

Practice writing the odd numbers from 1 to 1000.

Odd Numbers from 1 to 100

Odd Numbers

Odd Numbers from 101 to 200

101111121131141151161171181191
103113123133143153163173183193
105115125135145155165175185195
107117127137147157167177187197
109119129139149159169179189199

Odd Numbers from 201 to 300

201211221231241251261271281291
203213223233243253263273283293
205215225235245255265275285295
207217227237247257267277287297
209219229239249259269279289299

Odd Numbers from 301 to 400

301311321331341351351361371381
303313323333343353353363373383
305315325335345355355365375385
307317327337347357357367377387
309319329339349359359369379389

Odd Numbers from 401 to 500

401411421431441451461471481491
403413423433443453463473483493
405415425435445455465475485495
407417427437447457467477487497
409419429439449459469479489499

Properties

The 4 main properties are:

Adding 2 Odd Numbers

Any 2 odd numbers, when added, always gives an even number.

Odd number + Odd number = Even number

Proof:

Let a and b are 2 odd numbers

They are written in the form

a = 2k1 + 1, b = 2k2 + 1 where k1, k2 ∈ Z

Adding a and b, we get

(2k1 + 1) + (2k2 + 1)

=> 2k1 + 2k2 + 2

=> 2(k1 + k2 + 1)

This expression is divisible by 2s

Similarly, the sum of the first n odd numbers follow Sn = n2 rule.

Subtracting 2 Odd Numbers

When an odd number is subtracted from the other, it always gives an even number. It is similar to adding two odd numbers.

Odd Number – Odd number = Even number

Multiplication of 2 Odd Numbers

When an odd number is multiplied by another odd number, the product is also an odd number.

Odd number × Odd number = Odd number

Proof:

Let a and b are 2 odd numbers

a = 2k1 + 1, b = 2k2 + 1 where k1, k2 ∈ Z

Now, a × b = (2k1 + 1)(2k2 + 1)

=>4kk2 + 2k1 + 2k2 + 1

=>2(2k1 k2 + k1 + k2) + 1

This expression is an odd number

Division of 2 Odd Numbers

Division of 2 odd numbers always results in an odd number when the denominator is a factor of the numerator.

Odd number ÷ Odd number = Odd number

Types

There are 2 types of odd numbers. They are:

Consecutive Odd Numbers

They are 2 odd numbers that come one after the other in a sequence. If ‘a’ is an odd number, then the consecutive odd number corresponding to a is ‘a + 2’.They can be positive or negative.

Examples

Positive Consecutive Odd Number

  • 5 and 7
  • 11 and 13
  • 51 and 53
  • 101 and 103

Negative Consecutive Odd Number

  • -5 and -7
  • -11 and -13
  • -51 and -53

Composite Odd Number

They are positive odd numbers obtained by multiplying 2 smaller positive integers or multiplying the number with 1.

Given are the composite odd numbers till 100:

9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, and 99.

Thus, the smallest odd composite number is 9.

Solved Examples

Determine if 145 is an odd number or not.

Solution:

The given number is 145
Checking the divisibility of the number by 2, we get the remainder by 1, which proves that 145 is an odd number.

Simplify:
a) (-5) + (-9)
b) (-11) – (-3)
c) (-3) × (-7)
d) (-9) ÷ (-3)

Solution:

a) (-5) + (-9) = -14
b)(-11) – (-3) = -8
c) (-3) × (-7) = 21
d) (-9) ÷ (-3) = 3

The sum of three consecutive odd numbers is 51. Find the numbers.

Solution:

Let x be an odd number
Then the next consecutive odd number are x + 2 and the next term is x + 2 + 2 = x + 4
Now,
x + (x + 2) + (x + 4) = 3x + 6 = 51
=> 3x = 51 – 6
=> x = ${\dfrac{45}{3}}$ = 15
Hence, the other numbers are 15 + 2 = 17, 15 + 4 = 19
Thus the 3 consecutive odd numbers are 15, 17, and 19.

The sum of 5 consecutive odd numbers is 145. Find the third number in the sequence.

Solution:

Let x be an odd number
Then the next consecutive odd numbers are:
x + 2
x + 2 + 2 = x + 4
x + 2 + 2 + 2 = x + 6
x + 2 + 2 + 2 + 2 = x + 8
Now,
x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 5x + 20 = 145
=> 5x = 145 – 20
=> 5x = 125
=> x = ${\dfrac{125}{5}}$ = 25
Hence, the other numbers are
25 + 2 = 27
27 + 2 = 29
29 + 2 = 31
31 + 2 = 33
Thus, the third number in the sequence is 29.

Last modified on August 3rd, 2023

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