Hello everyone,
Trying to get my head around deductions and the deductive step using my text book, could someone look over my work:
Question:
n+6 is odd if and only if 5n+1 is even.
So my working, here it goes:
n+6=2k+1
n=2k-5
thus
5n+1=5(2k-5)+1
=2(5k-12)
so 5n+1 is even since 5k-12 is an integer.
5n+1=2k
subtract 4n and add +5 to both sides leaves.
n+6=2k-4n+5
=2(k-2n)+5
so n+2 is odd since k-2n is an integer.
Any of this making sense to anyone?
Thanks!
n+6 is odd if and only if 5n+1 is even
if 5n+1 is even, 5n is odd. So, n is odd (only odd*odd = odd)
If n is odd, n+6 is odd, (only odd+even=odd)
if n+6 is odd, n is odd, since odd numbers differ by 2.
so, 5n is odd, making 5n+1 even.
USING
odd*odd = odd since
(2m+1)(2n+1) = 2(2mn+m+n)+1 = 2k+1
even+odd = odd since
2m + 2n+1 = 2(m+n)+1 = 2k+1
I assume you are allowed to use the basic rules of addition and multiplication of odds and evens
odd + odd ---> even
odd + even --->odd
even + even ---> even
so if 5n + 1 is even: given
then 5n is odd: odd + 1 ---> even
odd*odd ---> odd
odd*even ---> even
even*even ---> even
so if 5n is odd, n has to be odd, since only odd*odd yields an odd
if n is odd, then n+1 is even, since
odd + odd---> even
my last 2 line should say:
if n is odd, then n+6 is odd, since
odd + even ---> odd
Hello!
Your work seems to be on the right track, but there is one mistake in your deductions:
You correctly started by assuming that n+6 is odd, which can be represented as n+6=2k+1, where k is an integer.
Then, you substituted n in terms of k, which is correct: n=2k-5.
However, when you calculated 5n+1, there is a small mistake:
5n+1 = 5(2k-5)+1
= 10k - 25 + 1
= 10k - 24
Now, you correctly observe that 5n+1 can be expressed as 2(5k - 12), which shows that it is even since it is divisible by 2.
Now, let's move on to the second part:
You correctly state that n+6 can also be expressed as n+6 = 2k+1.
Then, you rearrange the equation to n = 2k-5.
But when you start calculating n+6, there is another small error:
n+6 = 2k-5+6
= 2k+1
So, n+6 is actually 2k+1, not 2k-4n+5 as you wrote.
And since 2k+1 is odd, you correctly conclude that n+2 is odd because n+6 is odd.
To summarize:
- You correctly deduced that 5n+1 is even since it can be expressed as 2(5k - 12).
- However, there was a small mistake when calculating n+6, which should be 2k+1.
- Nonetheless, you correctly deduced that n+2 is odd because n+6 is odd.
I hope this clarifies the deductive steps for you! Let me know if you have any further questions.