91Ó°ÊÓ

Let \(a\) be a positive integer whose base- 10 representation is \(a=\) \(\left(a_{k-1} \cdots a_{1} a_{0}\right)_{10} .\) Let \(b\) be the sum of the decimal digits of \(a\); that is, let \(b:=\) \(a_{0}+a_{1}+\cdots+a_{k-1} .\) Show that \(a \equiv b(\bmod 9)\). From this, justify the usual "rules of thumb" for determining divisibility by 9 and \(3: a\) is divisible by 9 (respectively, 3\()\) if and only if the sum of the decimal digits of \(a\) is divisible by 9 (respectively, 3 ).

Find all elements of \(\mathbb{Z}_{19}^{*}\) of multiplicative order \(18 .\)

Let \(n=p q,\) where \(p\) and \(q\) are distinct primes. Show that if \(m:=\operatorname{lcm}(p-1, q-1),\) then \(\alpha^{m}=1\) for all \(\alpha \in \mathbb{Z}_{n}^{*}\).

Let \(p\) be any prime other than 2 or 5 . Show that \(p\) divides infinitely many of the numbers \(9,99,999,\) etc.

Let \(\tau(n)\) be the number of positive divisors of \(n .\) Show that: (a) \(\tau\) is a multiplicative function; (b) \(\tau(n)=\prod_{i=1}^{r}\left(e_{i}+1\right),\) where \(n=p_{1}^{e_{1}} \cdots p_{r}^{e_{r}}\) is the prime factorization of \(n\); (c) \(\sum_{d \mid n} \mu(d) \tau(n / d)=1\) (d) \(\sum_{d \mid n} \mu(d) \tau(d)=(-1)^{r},\) where \(n=p_{1}^{e_{1}} \cdots p_{r}^{e_{r}}\) is the prime factorization of \(n\).

The Mangoldt function \(\Lambda(n)\) is defined for all positive integers \(n\) as follows: \(\Lambda(n):=\log p,\) if \(n=p^{k}\) for some prime \(p\) and positive integer \(k,\) and \(\Lambda(n):=0,\) otherwise. Show that \(\sum_{d \mid n} \Lambda(d)=\log n,\) and from this, deduce that \(\Lambda(n)=-\sum_{d \mid n} \mu(d) \log d\).