Question

Show that any positive odd integer is of the from 6q+1, or 69 +3, or 69+5,where q is some integer.

NEET

Maths

Solution

901

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Let ( a ) be positive odd integer. By division algorithm [ a=6 times q+r ; 0 leq r<6 ] ( Rightarrow a=6 q+0, ) or ( 6 q+1 ) or ( quad 6 q+2, ) or ( 6 q+3 ) or ( quad 6 q+4, ) or ( 6 q+5 ) We reject ( 6 q+0,6 q+2, ) and ( 6 q+4 ) because they are even integers. Hence, the possible form of ( a ) is ( 6 q+1 ) or ( 6 q+3, ) or ( 6 q+5 )