91Ó°ÊÓ

(i) Prove that \(2^{n}>n^{3}\) for all \(n \geq 10\). (ii) Prove that \(2^{n}>n^{4}\) for all \(n \geq 17\). (iii) If \(k\) is a natural number, prove that \(2^{n}>n^{k}\) for all \(n \geq k^{2}+1\).

For any integer \(n \geq 2\), prove that there are \(n\) consecutive composite numbers. Conclude that the gap between consecutive primes can be arbitrarily large.

(Double Induction) Let \(S(m, n)\) be a doubly indexed family of statements, one for cach \(m \geq 0\) and \(n \geq 0 .\) Suppose that (i) \(S(0,0)\) is true; (ii) if \(S(m, 0)\) is true, then \(S(m+1,0)\) is true; (iii) if \(S(m, n)\) is true for all \(m\), then \(S(m, n+1)\) is true for all \(m\). Prove that \(S(m, n)\) is true for all \(m \geq 0\) and \(n \geq 0\)

(i) Show, for every \(n \geq 1\), that the "alternating sum" of the binomial coefficients is zero: $$ \left(\begin{array}{l} n \\ 0 \end{array}\right)-\left(\begin{array}{l} n \\ 1 \end{array}\right)+\left(\begin{array}{l} n \\ 2 \end{array}\right)-\cdots+(-1)^{n}\left(\begin{array}{l} n \\ n \end{array}\right)=0 $$

If \(a\) and \(b\) are relatively prime and if each divides an integer \(n\), then their product \(a b\) also divides \(n\)

Prove that a positive integer \(n\) is divisible by 11 if and only if the alternating sum of its digits is divisible by 11 (if the digits of \(a\) are \(d_{k} \ldots d_{2} d_{1} d_{0}\), then their alternating sum is \(\left.d_{0}-d_{1}+d_{2}-\cdots\right)\).

(i) Show that \(1000 \equiv-1 \bmod 7\). (ii) Show that if \(a=r_{0}+1000 r_{1}+1000^{2} r_{2}+\cdots\), then \(a\) is divisible by 7 if and only if \(r_{0}-r_{1}+r_{2}-\cdots\) is divisible by \(7 .\)

For a given positive integer \(m\), find all integers \(r\) with \(0

Prove that there is no perfect square \(a^{2}\) whose last two digits are 35 .

How many times in 1900 did the first day of the month fall on a Tuesday?