Answer and Step-by-step explanation:
Solution:
Given:
m, n, and p are integers
m + n and n + p are odd integers.
Then prove that m + p is even integer.
Proof:
Properties of odd and even integer:
If x is an odd integer, then exist an integer y such that x = 2y + 1.
If x is an even integer, then there exist an integer y such that x = 2y.
Let m + n and n + p be odd integers, then there exist integer y and z such that :
m + n = 2y + 1
and n + p = 2z + 1
let add the two odd integers:
m + n+ n + p = 2y + 1 + 2z + 1
m + 2n + p = 2y + 2z + 2
we are interested in m + p , so subtract 2n from both sides:
m + 2n – 2n + p = 2y + 2z+ 2 – 2n
m + p = 2y + 2z – 2n -2
by taking common 2 from right side:
m + p = 2 ( y + z – n + 1)
since y, z and n are integers, y + z – n is also an integer and thus m + p is an even integer.
Hence proved if m, n, and p are integers and m + n and n + p are odd integers. Then, m + p is even integer..