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:

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 from 101 to 200

101	111	121	131	141	151	161	171	181	191
103	113	123	133	143	153	163	173	183	193
105	115	125	135	145	155	165	175	185	195
107	117	127	137	147	157	167	177	187	197
109	119	129	139	149	159	169	179	189	199

Odd Numbers from 201 to 300

201	211	221	231	241	251	261	271	281	291
203	213	223	233	243	253	263	273	283	293
205	215	225	235	245	255	265	275	285	295
207	217	227	237	247	257	267	277	287	297
209	219	229	239	249	259	269	279	289	299

Odd Numbers from 301 to 400

301	311	321	331	341	351	351	361	371	381
303	313	323	333	343	353	353	363	373	383
305	315	325	335	345	355	355	365	375	385
307	317	327	337	347	357	357	367	377	387
309	319	329	339	349	359	359	369	379	389

Odd Numbers from 401 to 500

401	411	421	431	441	451	461	471	481	491
403	413	423	433	443	453	463	473	483	493
405	415	425	435	445	455	465	475	485	495
407	417	427	437	447	457	467	477	487	497
409	419	429	439	449	459	469	479	489	499

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 S_n = 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)

=>4k₁ k₂ + 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