Assignment 1.Prove the statement. a, There are distinct integers m and n such th
ID: 3142410 • Letter: A
Question
Assignment
1.Prove the statement.
a, There are distinct integers m and n such that 1 m + 1 n is an integer.
b, There are real numbers a and b such that a + b = a + b.
2. Prove the statements.
a, The sum of any two odd integers is even.
b, For all integers n, if n is odd then n 2 is odd.
3. Determine whether the statement is true or false. Justify your answer with a proof or a counterexample.
1, The negative of any odd integer is odd.
2, The different of ant two odd integers is odd.
3, For all integers n and m, if n m is even then n 3 m3 is even.
4, For all integers n, n 2 n + 11 is prime number.
Explanation / Answer
(According to Chegg policy, only four subquestions will be answered. Please post the remaining in another question)
1. a. 1 m + 1 n = m+n
If m n, the result is automatically proved,
If m = n, then m+1 n-1 and m+1 + n-1 = m+n.
Thus the result is proved.
b. Let a = 0
=> a = 0
=> (ab) = 0
=> 2(ab) = 0
=> a + b + 2(ab) = a + b
=> (a + b)2 = (a+b)2
=> a + b = (a+b)
2. a. Let a and b be two odd integers.
Let a = 2k+1 and b = 2l+1 where k and l are integers
=> a + b = 2k+1 + 2l+1
=> a + b = 2k+2l+2
=> a + b = 2(k+l+1)
Since RHS has a factor 2, LHS must be divisible by 2
=> a + b is divisible by 2
=> a + b is even.
b. Since n is odd, let n = 2k+1 where k is an integer.
Squaring, n2 = (2k+1)2
=> n2 = 4k2 + 4k + 1
=> n2 = 2(2k2 + 2k) + 1
Let 2k2 + 2k = l
=> n2 = 2l + 1
Since n2 is of the form 2l+1, n2 is odd.