Practice (91)

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Find the number $x = [1, 2, 3, 1, 2, 3, \cdots]$. (continued fraction)

Let $n$ be a positive integer that is one less than a multiple of 24. Prove that if $a$ and $b$ are positive integers such that $ab=n$, then $a+b$ is a multiple of 24.

Find the smallest positive integer $n$ such that the remainder is always $1$ when $n$ is divided by $2$, $3$, $4$, $5$, or $6$. In addition, $n$ must be a multiple of $7$.

Show that any positive integer can be expressed as a sum of integers which are some power of 3.

In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers? Regard $m\cdot n$ and $n\cdot m$ as the same.

A binary palindrome is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same backward or forward). (Leading zeros are not permitted in the standard representation.) For example, 2015 is a binary palindrome, because in base 2 it is 11111011111. How many positive integers less than 2015 are binary palindromes?

In baseball, a player's batting average is the number of hits divided by the number of at bats, rounded to three decimal places. Danielle's batting average is $0.399$. What is the fewest number of at bets that Danielle could have?

Show that if $k \ge 4$, then $lcm(1; 3;\cdots; 2k- 3; 2k- 1) > (2k + 1)^2$ where $lcm$ stands for least common multiple.

Find all positive integers $n$ such that for all odd integers $a$. If $a^2\le n$, then $a|n$.

Find all $n \in \mathbb{Z}^+$ such that $2^n + n | 8^n + n$.

Find all nonnegative integers $n$ such that there are integers $a$ and $b$ with the property: $$n^2 = a + b \qquad\text{and}\qquad n^3 = a^2 + b^2$$

Prove that $2^n + 1$ has no prime factors of the form $8k + 7$.

Find all triples $(a; b; c)$ of natural numbers such that $lcm(a; b; c) = a + b + c$.

Find all natural numbers $n$ such that $n$ equals the cube of the sum of its digits.

Find all odd integers $n$ for which $n|3^n + 1$.

Let $p$ be a prime and $a, b, c \in \mathbb{Z}^+$, such that $p = a+b+c-1$ and $p|a^3+b^3+c^3-1$. Prove that $min (a, b, c) = 1$

Find all primes $p, q$ such that $pq | 2^p + 2^q$.

Let $A = 6^n$ for real $n$. Find all natural numbers $n$ such that $n^{A+2} + n^{A+1} + 1$ is a prime number.

Let $a$, $b$, and $c$ be three odd integers. Prove the equation $ax^2 + bx + c=0$ does not have rational roots.

Show that the difference of two squares of odd numbers must be a multiple of $8$.

Find the least positive integer $n$ such that for every prime number $p$, $p^2 + n$ is never prime.

Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 prime numbers?

A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?

Let $a$, $b$, and $c$ form a geometric sequence. Can the last two digits of $N=a^3+b^3+c^3-3abc$ be 20?

Show that there exists an infinite sequence of positive integers $a_1, a_2, \cdots$ such that $$S_n=a_1^2 + a_2^2 + \cdots + a_n^2$$ is square for any positive integer $n$.