Question

Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5 ,where a is some integet

Answer 1

Euclid division lemma:

two positive integers “a” and “b”, then there exists unique integers “q” and “r” such that which satisfies the condition a = bq + r where 0 ≤ r ≤ b.

Step-by-step explanation:

a=6q+1

where b=6; 0 ≤ r ≤ 6

The possible remainders are 0,1,2,3,4,5

a=6q+0=6q

a=6q+1

a=6q+2

a=6q+3

a=6q+4

a=6q+5

where q is quotient.....

By the problem a is odd integer.....

therefore, a cannot be 6q,6q+2,6q+4......

Hence, Any odd integer is of the form 6q+1 or 6q+3 or 6q+5......

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