Practice (90)
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For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice.
The numbers from 1 to 7 are separated into two non-empty sets A and B. The numbers in A are multiplied together to get a. The numbers in B are multiplied together to get b. The larger of the two numbers a and b is written down. What is the smallest number that can be written down using this procedure?
The sum of three distinct 2-digit primes is 53. Two of the primes have a units digit of 3, and the other prime has a units digit of 7. What is the greatest of the three primes?
If $2016$ consecutive integers are added together, where the $999^{th}$ number in the sequence is $1,244,584$, what is the remainder when this sum is divided by $6$?
If 738 consecutive integers are added together, where the 178th number in the sequence is 4,256,815, what is the remainder when this sum is divided by 6?
Find all positive integer $n$ such that $n^2 + 2^n$ is a perfect square.
Given that $a, b,$ and $c$ are positive integers such that $a^b\cdot b^c$ is a multiple of 2016. Compute the least possible value of $a+b+c$.
Find the largest of three prime divisors of $13^4+16^5-172^2$.
In $\triangle{ABC}$, $AB = 33, AC=21,$ and $BC=m$ where $m$ is an integer. There exist points $D$ and $E$ on $AB$ and $AC$, respectively, such that $AD=DE=EC=n$ where $n$ is also an integer. Find all the possible values of $m$.
Prove that $7\mid 8^n-1$ for $n\ge 1$.
Show that $5\mid 4^{2n}-1$ for $n\ge 1$.
Prove that $15\mid 4^{2n}-1$ for $n\ge 1$.
Prove that for every positive integer $n$, there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Which positive integer?
Let $S_n$ be the minimal value of $\displaystyle\sum_{k=1}^n\sqrt{a_k^2+b_k^2}$ where $\{a_k\}$ is an arithmetic sequence whose first term is $4$ and common difference is $8$. $b_1, b_2,\cdots, b_n$ are positive real numbers satisfying $\displaystyle\sum_{k=1}^nb_k=17$. If there exist a positive integer $n$ such that $S_n$ is also an integer, find $n$.
The number $N$ is a two-digit number.
• When $N$ is divided by $9$, the remainder is $1$.
• When $N$ is divided by $10$, the remainder is $3$.
What is the remainder when $N$ is divided by $11$?
What is the sum of the distinct prime integer divisors of $2016$?
The sum of two prime numbers is 85. What is the product of these two prime numbers?
If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?
The 7-digit numbers $\underline{7} \underline{4} \underline{A} \underline{5} \underline{2} \underline{B} \underline{1}$ and $\underline{3} \underline{2} \underline{6} \underline{A} \underline{B} \underline{4} \underline{C}$ are each multiples of 3. Which of the following could be the value of $C$?
Find all postive integers $(a,b,c)$ such that
$$ab-c,\quad bc-a,\quad ca-b$$are all powers of $2$.
Proposed by Serbia
Show that if $n$ is an integer greater than $1$, then $(2^n-1)$ is not divisible by $n$.
Show that every integer $k > 1$ has a multiple which is less than $k^4$ and can be written in base 10 using at most 4 different digits.
In a sports contest, there were $m$ medals awarded on $n$ successive days ($n > 1$). On the first day, one medal and $1/7$ of the remaining $m − 1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining
medals were awarded; and so on. On the $n^{th}$ and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?
If all roots of the equation $$x^4-16x^3+(81-2a)x^2 +(16a-142)x+(a^2-21a+68)=0$$ are integers, find the value of $a$ and solve this equation.