Show that any positive odd integer is of the form 6q+1, or 6q+3, or 6q+5, where q is some integer.
Show that any positive odd integer is of the form 6q+1, or 6q+3, or 6q+5, where q is some integer.
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Sol:-
Let us start with taking a, where a is a positive odd integer. We apply the division algorithm with a and b = 6.
Since 0 ≤ r < 6, the possible remainders are 0, 1, 2 , 3, 4 and 5.
That is, a can be 6q, or 6q + 1, or 6q + 2, or 6q + 3 or 6q + 4 or 6q + 5, where q is the quotient.
However, since a is odd, a cannot be 6q or 6q + 2 or 6q + 4 (since they are divisible by 2). Therefore, any odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5.