**This is the second note in the series Number Theory: Divisibility. All numbers involved in this note are integers, and letters used in this note stand for integers without further specification.**

**Recap of Properties:**

$(1)$ If $b \mid c,$ and $c \mid a,$ then $b \mid a,$ that is, divisibility is transitive.

$(2)$ If $b \mid a,$ and $b \mid c,$ then $b \mid (a \pm c),$ that is, the set of multiples of an integer is closed under addition and subtraction operations.

By using this property repeatedly, we have, if $b \mid a$ and $b \mid c$, then $b \mid (au+cv)$, for any integers $u$ and $v$. In general, if $a_1,a_2,\cdots,a_n$ are multiples of $b$, then $b \mid (a_1+a_2+\cdots+a_n).$

$(3)$ If $b \mid a$, then $a=0$ or $\lvert a \rvert \geq \lvert b \rvert$. Thus, if $b \mid a$ and $a \mid b$, then $\lvert a \rvert = \lvert b \rvert$.

Clearly, for any two integers $a$ and $b$, $a$ is not always divisible by $b$. But we have the following result, which is called the division algorithm. It is the most important result in elementary number theory.

$(4)$ (The division algorithm) Let $a$ and $b$ be integers, and $b > 0$. Then there is a unique pair of integers $q$ and $r$, such that $a=bq+r \hspace{2mm} \text{and} \hspace{2mm} 0 \leq r < b.$

The integer $q$ is called the (incomplete)

quotientwhen $a$ is divided by $b$, $r$ called theremainder. Note that the values of $r$ has $b$ kinds of possibilities, $0,1,\cdots,b-1$. If $r=0$, then $a$ is divisible by $b$.It is easy to see that the quotient $q$ in the division algorithm is in fact $\lfloor\frac{a}{b}\rfloor$ (the greatest integer not exceeding $\frac{a}{b}$), and the heart of the division algorithm is the inequality about the remainder $r$: $0 \leq r < b$. We will go back to this point later on.

The basic method of proving $b \mid a$ is to factorize $a$ into the product of $b$ and another integer. Usually, in some basic problems this kind of factorization can be obtained by taking some special value in algebraic factorization equations. The following two factorization formulae are very useful in proving this kind of problems.

$(5)$ If $n$ is a positive integer, then $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\cdots+xy^{n-2}+y^{n-1}).$

$(6)$ If $n$ is a positive odd number, then $x^n+y^n=(x+y)(x^{n-1}-x^{n-2}y+\cdots-xy^{n-2}+y^{n-1}).$

**Examples:**

Example 1:Prove that $1\underbrace{0\cdots 0}_{200}1$ is divisible by $1001$.

Proof:By $(6)$, we have$1\underbrace{0\cdots 0}_{200}1=10^{201}+1=(10^3)^{67}+1$

$=(10^3+1)\left[(10^3)^{66}-(10^3)^{65}+\cdots -10^3+1\right]$

$=1001\left[(10^3)^{66}-(10^3)^{65}+\cdots -10^3+1\right].$

Therefore, $1001$ divides $1\underbrace{0\cdots 0}_{200}1$.

Example 2:Let $a$ and $b$ be integers such that $a>b\geq 0$. Show that$(2^{2^b}+1)\mid (2^{2^a}-1).$

Proof:Take $x=2^{2^{b+1}}$, $y=1$ in $(5)$, and substitute $n$ by $2^{a-b-1}$. We have$2^{2^a}-1=(2^{2^{b+1}}-1)\left[(2^{2^{b+1}})^{2^{a-b-1}-1}+\cdots+2^{2^{b+1}}+1\right].$

Thus,

$(2^{2^{b+1}}-1)\mid (2^{2^a}-1).$

Since

$2^{2^{b+1}}-1=(2^{2^b}-1)(2^{2^b}+1),$

hence

$(2^{2^b}+1)\mid (2^{2^{b+1}}-1).$

By $(1)$, we have

$(2^{2^b}+1)\mid(2^{2^a}-1),$

and we are done.

Example 3:For a positive integer $n$, let $S(n)$ denote the sum of digits of $n$. Show that $9\mid n$ if and only if $9\mid S(n)$.

Proof:Write $n=a_k\times 10^k +\cdots + a_1\times 10 + a_0$ (where $0\leq a_i \leq 9$, and $a_k \neq 0$), then $S(n)=a_0+a_1+\cdots+a_k$. We have$n-S(n)=a_k(10^k-1)+\cdots+a_1(10-1)\cdots(1.1)$

For $1 \leq i \leq k$, from $(5)$ we get $9 \mid (10^i-1)$. So every term of the $k$ terms in the RHS of $(1.1)$ is a multiple of $9$, thus $(2)$ implies that their sum is also a multiple of $9$, that is, $9 \mid (n-S(n))$. Hence, the result can be obtained easily.

Example 4:Let $k \geq 1$ be an odd integer. Prove that for any positive integer $n$, $1^k+2^k+\cdots+n^k$ is not divisible by $n+2$.

Proof:When $n=1$ the statement is obviously true. For $n \geq 2$, denote the sum by $A$. Then$2A=2+(2^k+n^k)+(3^k+(n-1)^k)+\cdots+(n^k+2^k).$

Since $k$ is a positive odd integer, from $(6)$ we know that for every $i \geq 2$, $i^k+(n+2-i)^k$ is divisible by

$i+(n+2-i)=n+2.$

Thus $2A$ has remainder $2$ when divided by $n+2$, which implies that $A$ is not divisible by $n+2$ (note that $n+2>2$).

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