show that any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5 where q is some integer
Answers
Answer:
According to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition a = bq + r where 0 ≤ r < b.
Let a be the positive odd integer which when divided by 6 gives q as quotient and r as remainder.
According to Euclid’s division lemma
a = bq + r
a = 6q + r………………….(1)
where, (0 ≤ r < 6)
So r can be either 0, 1, 2, 3, 4 and 5.
Case 1:
If r = 1, then equation (1) becomes
a = 6q + 1
The Above equation will be always an odd integer.
Case 2:
If r = 3, then equation (1) becomes
a = 6q + 3
The Above equation will be always an odd integer.
Case 3:
If r = 5, then equation (1) becomes
a = 6q + 5
The above equation will be always an odd integer.
∴ Any odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5.
Hence proved.
Check out the video below for more information on Euclid’s division algorithm
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3 Comments
Vansh MAY 18, 2021 AT 6:32 AM
nice