Show that any positive odd integer is of the form 6q+1, 6q+3 or 6q+5, where q is any positive integer
Show that any positive odd integer is of the form 6q+1,or 6q+3 or 6q+3 or 6q+5 where is some integer
The odd numbers 1,3, and 5 cannot be obtained using any of the above expressions with q a positive integer.
Only if q = 0, and q is not a positive integer.
To show that any positive odd integer can be expressed in the form 6q+1, 6q+3, or 6q+5, we can consider the possible remainders when dividing an odd integer by 6.
When dividing any positive integer by 6, the remainder can be one of the following: 0, 1, 2, 3, 4, or 5.
Let's examine each remainder case:
1) Remainder 0: If an odd integer leaves a remainder of 0 when divided by 6, then it can be expressed as 6q, where q is a positive integer. However, this implies that the number is divisible by 6, and hence, it cannot be an odd integer.
2) Remainder 1: If an odd integer leaves a remainder of 1 when divided by 6, then we can express it as 6q+1. Here, q is the quotient obtained after dividing the given odd integer by 6.
3) Remainder 2: If an odd integer leaves a remainder of 2 when divided by 6, then it can be expressed as 6q+2. However, in this case, the number is even and not odd.
4) Remainder 3: If an odd integer leaves a remainder of 3 when divided by 6, then we can express it as 6q+3.
5) Remainder 4: If an odd integer leaves a remainder of 4 when divided by 6, then it can be expressed as 6q+4. However, like in the previous case, the number is even and not odd.
6) Remainder 5: If an odd integer leaves a remainder of 5 when divided by 6, then we can express it as 6q+5.
Hence, any positive odd integer can be written in the form 6q+1, 6q+3, or 6q+5, where q is any positive integer.