Question 1192006: Prove:
If m and n are integers, then 4 is divisible by (m + n)^2 - (m-n)^2
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
(m + n)^2 = (m + n)(m + n)
(m + n)^2 = m(m + n) + n(m + n)
(m + n)^2 = m^2 + mn + mn + n^2
(m + n)^2 = m^2 + 2mn + n^2
(m - n)^2 = (m - n)(m - n)
(m - n)^2 = m(m - n) - n(m - n)
(m - n)^2 = m^2 - mn - mn + n^2
(m - n)^2 = m^2 - 2mn + n^2
Or you can use this route
(m + n)^2 = m^2 + 2mn + n^2
(m + (-n))^2 = m^2 + 2m(-n) + (-n)^2
(m - n)^2 = m^2 - 2mn + n^2
Now let's subtract the two expressions
(m + n)^2 - (m - n)^2
(m^2 + 2mn + n^2) - (m^2 - 2mn + n^2)
m^2 + 2mn + n^2 - m^2 + 2mn - n^2
(m^2-m^2) + (2mn+2mn) + (n^2 - n^2)
0m^2 + 4mn + 0n^2
0 + 4mn + 0
4mn
Therefore,
(m + n)^2 - (m - n)^2 = 4mn
is an identity for any real numbers m,n.
Since 4 is a factor of 4mn, this proves that 4mn is divisible by 4.
Consequently, (m + n)^2 - (m - n)^2 is also divisible by 4.
This concludes the proof.
The phrasing "4 is divisible by (m + n)^2 - (m - n)^2" is not correct because the order should be swapped.
If your teacher meant "4 is a factor of (m + n)^2 - (m - n)^2", then it would be the correct terminology.
Another way to phrase it would be to say "(m+n)^2 - (m-n)^2 is a multiple of 4".
Or you could write "4 divides (m + n)^2 - (m - n)^2".
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