Question

use Euclid division lemma to show that any positive odd integers is of the form 6q+1, or 6q+3 or 6q+5 where q is some integers ​

Answer 1

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Answer 2

Solution :-

Let a be any odd positive integer and b=6

By using Euclid’s algorithm :

• a = 6q + r { 0 ≤ r < 6 }

• Possibilities of "r" :- {0,1,2,3,4,5}

Case - I , When r = 0

• a = 6q + 0 = 6q

Case - 2 , When r = 1

• a = 6q + 1 ...... (1)

Case - 3, When r = 2

• a = 6q + 2

Case - 4, when r = 3

• a = 6q + 3 ...... (2)

Case - 5, when r = 4

• a = 6q + 4

Case - 6, when r = 5

• a = 6q + 5 ....... (3)

Therefore, positive odd integers is of the form 6q+1, or 6q+3 or 6q+5