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Show that. Any positive odd integer …

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Show that. Any positive odd integer is form of 6q+1or6q+3or6q+5 whereq is some integer

Posted by Sujit Waghmare 6 years, 5 months ago

CBSE > Class 10 > Mathematics

3 answers

Krishna Kant Pandey 6 years, 5 months ago

&

Sia ? 6 years, 5 months ago

Let a be any positive integer and b = 6
∴ by Euclid’s division lemma
a = bq + r, 0≤ r and q be any integer q ≥ 0
∴ a = 6q + r,
where, r = 0, 1, 2, 3, 4, 5
If a is even then then remainder by division of 6 is 0,2 or 4

Hence r=0,2,or 4

or A is of form 6q,6q+2,6q+4
As, a = 6q = 2(3q), or
a = 6q + 2 = 2(3q + 1), or
a = 6q + 4 = 2(3q + 2).
If these 3 cases a is an even integer.
but if the remainder is 1,3 or 5 then r=1,3 or 5

or A is of form 6q+1,,6q+3 or,6q+5
Case 1:a = 6q + 1 = 2(3q) + 1 = 2n + 1,
Case 2: a = 6q + 3 = 6q + 2 + 1,
= 2(3q + 1) + 1 = 2n + 1,
Case 3: a = 6q + 5 = 6q + 4 + 1
= 2(3q + 2) + 1 = 2n + 1

This shows that odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

Aviral Pal 6 years, 5 months ago

A=bq+r Let b=6 and r= 0,1,2,3,4,5 Putting r=0 A=6q Putting r=1 A=6q+1 Putting r=2 6q+2 Putting r=3 6q+3 Putting r=4 6q+4 Putting r=5 6q+5 We need positive odd integer so 6q+1,6q+3, 6q+5 are the positive odd integer

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