How to prove that any positive odd number is in the form of 6q+1, 6q+3, 6q+5?
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we know an odd number is represented by
2k+1 where k is an integer------------0
so 6q+1 must be equal to 2k+1
I. E. 6q+1=2k+1
=>6q=2k
=>k=3q ---------1
similarly 6q+3=2k+1
=>6q+2=2k
=>k=3q+1---------2
also 6q+5=2k+1
=>6q+4=2k
=>k=3q+2----------3
from 1 ,2,3 we get k is always an integer which satisfies our condition(0)
so the statement is true
2k+1 where k is an integer------------0
so 6q+1 must be equal to 2k+1
I. E. 6q+1=2k+1
=>6q=2k
=>k=3q ---------1
similarly 6q+3=2k+1
=>6q+2=2k
=>k=3q+1---------2
also 6q+5=2k+1
=>6q+4=2k
=>k=3q+2----------3
from 1 ,2,3 we get k is always an integer which satisfies our condition(0)
so the statement is true
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