Table of Contents

Last modified on March 28th, 2023

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.

Practice writing the odd numbers from 1 to 1000.

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 |

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 |

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 |

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 |

The 4 main properties are:

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 = 2k_{1} + 1, b = 2k_{2} + 1 where k_{1}, k_{2} ∈ Z

Adding a and b, we get

(2k_{1} + 1) + (2k_{2} + 1)

=> 2k_{1} + 2k_{2} + 2

=> 2(k_{1} + k_{2} + 1)

This expression is divisible by 2s

Similarly, the sum of the first n odd numbers follow S_{n} = n^{2} rule.

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

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 = 2k_{1} + 1, b = 2k_{2} + 1 where k_{1}, k_{2} ∈ Z

Now, a × b = (2k_{1} + 1)(2k_{2} + 1)

=>4k_{1 }k_{2} + 2k_{1} + 2k_{2} + 1

=>2(2k_{1} k_{2} + k_{1} + k_{2}) + 1

This expression is an odd number

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

There are 2 types of odd numbers. They are:

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

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.

**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 March 28th, 2023