problem archive
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Conjecture a formula for $\sum_{k=1}^{n} \frac{1}{k(k+1)}=\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\cdots+\frac{1}{n(n+1)}$ -
Show that any amount of postage that is an integer number of cents greater than 53 cents can be formed using just 7-cent and 10-cent stamps. -
Show by mathematical induction that if $h \geq-1$, then $1+n h \leq(1+h)^{n}$ for all nonnegative integers $n$. -
Explain what is wrong with the following proof by mathematical induction. -
Show that if $a$ and $b$ are positive integers, then there are unique integers $q$ and $r$ such that $a=b q+r$, where $-b / 2<r \leq b / 2$. This result is called the modified division algorithm. -
Show that if $a$ is an integer, then 3 divides $a^{3}-a$. -
Show that every nonzero integer can be uniquely represented in the form $$ e_{k} 3^{k}+e_{k-1} 3^{k-1}+\cdots+e_{1} 3+e_{0} $$ -
Consider a balance scale with 2 pans, $A$ and $B$. -
If the base $b$ expansion of $n$ is $n=\left(a_{k} a_{k-1} \ldots a_{1} a_{0}\right)_{b}$, what is the base $b$ expansion of $b^{m} n$ ? -
Show that no integer of the form $n^{3}+1$ is a prime, other than $2=1^{3}+1$. -
This exercise constructs another proof of the infinitude of primes. Show that the integer $Q_{n}=n !+1$, where $n$ is a positive integer, has a prime divisor greater than $n$. Conclude that there are infinitely many primes. -
Can you show that there are infinitely many primes by looking at the integers $S_{n}=n !-1$, where $n$ is a positive integer? -
et $a$ be a positive integer. What is the greatest common divisor of $a$ and $a+2$ ? -
Show that if $a$ and $b$ are integers with $(a, b)=1$, then $(a+b, a-b)=1$ or 2 . -
Show that if $a$ and $b$ are both even integers that are not both 0 , then $(a, b)=$ $2(a / 2, b / 2)$. -
Show that if $k$ is a positive integer, then $3 k+2$ and $5 k+3$ are relatively prime. -
Use the Euclidean algorithm to find each of the following greatest common divisors. -
Find the greatest common divisor of each of the following sets of integers. -
Show that if $a$ and $b$ are positive integers and $a^{3} \mid b^{2}$, then $a \mid b$. -
Which pairs of integers $a$ and $b$ have greatest common divisor 18 and least common multiple 540?