91Ó°ÊÓ

Prove each of the following for all \(n \geq 1\) by the Principle of Mathematical Induction. a) \(1^{2}+3^{2}+5^{2}+\cdots+(2 n-1)^{2}=\frac{n(2 n-1)(2 n+1)}{3}\) b) \(1 \cdot 3+2 \cdot 4+3 \cdot 5+\cdots+n(n+2)=\) \(\frac{n(n+1)(2 n+7)}{6}\) c) \(\sum_{i=1}^{n} \frac{1}{i(i+1)}=\frac{n}{n+1}\) d) \(\sum_{i=1}^{n} i^{3}=\frac{n^{2}(n+1)^{2}}{4}=\left(\sum_{l=1}^{n} i\right)^{2}\)

For each of the following pairs \(a, b \in \mathbf{Z}^{+}\), determine \(\operatorname{gcd}(a, b)\) and express it as a linear combination of \(a, b\). a) 231,1820 b) 1369,2597 c) 2689,4001

In the following pseudocode program segment the variables \(n\) and sum are integer variables. Following the execution of this program segment, which value of \(n\) is printed? \(n:=3\) sum \(:=0\) while sum \(<10,000\) do begin \(\quad n:=n+7\) \(\quad\) sum \(:=\) sum \(+n\) end print n

For \(a, b \in \mathbf{Z}^{+}\)and \(s, t \in \mathbf{Z}\), what can we say about \(\operatorname{gcd}(a, b)\) if a) \(a s+b t=2 ?\) b) \(a s+b t=3 ?\) c) \(a s+b t=4 ?\) d) \(a s+b t=6 ?\)

A wheel of fortune has the integers from 1 to 25 placed on it in a random manner. Show that regardless of how the numbers are positioned on the wheel, there are three adjacent numbers whose sum is at least 39 .

For \(a, b, n \in \mathbf{Z}^{+}\), prove that \(\operatorname{gcd}(n a, n b)=n \operatorname{gcd}(a, b)\).

Prove that \(\sqrt{p}\) is irrational for any prime \(p\).

Let \(n \in \mathbf{Z}^{+}\) a) Prove that \(\operatorname{gcd}(n, n+2)=1\) or 2 . b) What possible values can \(\operatorname{gcd}(n, n+3)\) have? What about \(\operatorname{gcd}(n, n+4) ?\) c) If \(k \in \mathbf{Z}^{+}\), what can we say about \(\operatorname{gcd}(n, n+k)\) ?

a) For the four-digit integers (from 1000 to 9999) how many are palindromes and what is their sum? b) Write a computer program to check the answer for the sum in part (a).

Let \(a, b, c \in \mathbf{Z}^{+}\)with \(\operatorname{gcd}(a, b)=1\). If \(a \mid b c\), prove that \(a \mid c\)