COSC 522
HW4
Due: 04/08/08
1. What proof strategy fits the best to prove that “if x is an integer divisible 6 and y is an integer
divisible by 4, then xy is an integer divisible by 12.” Why does it fit the best? Use your chosen
strategy to prove the above statement. (3 pts)
2. Prove that the following is true for all positive integers n: n is even if and only if 3n2  8 is even.
(Notice that it says if and only if. So you have to prove both ways)
3.
4.
5.
6.
7.
8.
9.
10.
11.
Show that if n is an integer and n3 + 5 is odd, then n is even.
Give a proof by contradiction of the following: If x and y are even integers, then xy is even.
Prove that if x is irrational then 1/x is irrational. (Hint: contraposition)
Give a proof by contradiction that if n is an integer and 3n+2 is even, then n is even.
Prove that the square of an even number is divisible by 4.
Prove that the sum of an integer and its square is even.
Show that at least 3 of any 25 days chosen must fall on the same month of the year.
Prove that n is odd if and only if 5n + 6 is odd.
Suppose you wish to prove that the following is true for all positive integers n by using the Principle
of Mathematical Induction: 1  4  7  10    (3n  2) 
n(3n  1)
.
2
a.
b.
c.
d.
Write and prove P(1)
Write P(72)
Write P(73)
Use the result from P(72) to find the sum 1 + 4 +7 +… +(3*73-2). Is your result the same as
73(3*73-1)/2?
e. Write P(k)
f. Write P(k
g. Use your result from P(k) to prove P(k+1).
h. Write a recursive function to compute the sum of sequence.
12. Suppose you wish to prove that the following is true for all positive integers n by using the Principle
of Mathematical Induction: 2  4  6    2n  n(n+1).
a. Write and prove P(1)
b. Write P(72)
c. Write P(73)
d. Use the result from P(72) to find the sum 2  4  6    2*73. Is your result the same as
73*(73+1)?
e. Write P(k)
f. Write P(k
g. Use your result from P(k) to prove P(k+1).
h. Write a recursive function to compute the sum of sequence.
13. Prove that the following is true for all positive integers n by using the Principle of Mathematical
Induction: 2  6  10    (4n-2)  2n2.
Write a recursive function to compute the sum of sequence.
COSC 522
HW4
Due: 04/08/08
14. Prove that the following is true for all positive integers n by using the Principle of Mathematical
Induction: 13  2 3  33    n 3 
n 2 (n  1) 2
. Write a recursive function for it.
4
Write a recursive function to compute the sum of sequence.
15. Use the Principle of Mathematical Induction to prove that 2  (n2  3n) for all n  1.
16. Prove that 23n -1 is divisible by 7 for all positive integers n by using the Principle of Mathematical
Induction.
17. Prove that the following is true for all positive integers n by using the Principle of Mathematical
2
2
2
2
Induction: 1  2  3    n 
n(2n  1)( 2n  1)
3
Write a recursive function to compute the sum of sequence.
18.
Prove that any amount of postage greater than or equal to 14 cents can be build using only 3 cent
and 8 cent stamps.
19.
Prove that a set with n elements has n(n-1)/2 subsets containing exactly two elements whenever n
is an integer greater than or equal to 2.
20.
Prove that a set with n elements has n(n-1)(n-2)/6 subsets containing exactly three elements
whenever n is an integer greater than or equal to 3. (Hint: you can use the above theorem from
problem 19 where necessary)