Found problems: 16
For every positive integer $n$ let $\tau (n)$ denote the number of its positive factors. Determine all $n \in N$ that satisfy the equality $\tau (n) = \frac{n}{3}$
A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?
Find all the natural numbers $N$ which satisfy the following properties:
(i) $N$ has exactly $6$ distinct factors $1, d_1, d_2, d_3, d_4, N$ and
(ii) $1 + N = 5(d_1 + d_2+d_3 + d_4)$.
Justify your answers.
Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.
Factor the polynomial $P (x) = 1 + x +x^2+...+x^{2^k-1}$
Prove that the polynomial $x^{200} y^{200} +1$ cannot be represented in the form $f(x)g(y)$, where $f$ and $g$ are polynomials of only $x$ and $y$, respectively.
The set of numbers $M=\{4k-3 | k\in N\}$ is considered. A number of of this set is called “simple” if it is impossible to put in the form of a product of numbers from $M$ other than $1$. Show that in this set, the decomposition of numbers in the product of "simple" factors is ambiguous.
The number $201212200619$ has a factor $m$ such that $6 \cdot 10^9 <m <6.5 \cdot 10^9$. Find $m$.
Let us call the number of factors in the prime decomposition of an integer $n > 1$ the complexity of $n$. For example, [i]complexity [/i] of numbers $4$ and $6$ is equal to $2$. Find all $n$ such that all integers between $n$ and $2n$ have complexity
a) not greater than the complexity of $n$.
b) less than the complexity of $n$.
(Boris Frenkin)
Determine the largest integer $k$ for which $12^k$ is a factor of $120! $
Show that there are infinitely many positive integers $n$ such that the number of distinct odd prime factors of $n(n + 3)$ is a multiple of $3$.
(For instance, $180 = 2^2 \cdot 3^2 \cdot 5$ has two distinct odd prime factors and $840 = 2^3 \cdot 3 \cdot 5 \cdot 7$ has three.)
Factor $(b - c)^3 + (c - a)^3 + (a - b)^3$.
The ${\textit{distinct prime factors}}$ of an integer are its prime factors listed without repetition. For example, the distinct prime factors of $40$ are $2$ and $5$.
Let $A=2^k - 2$ and $B= 2^k \cdot A$, where $k$ is an integer ($k \ge 2$).
Show that, for every integer $k$ greater than or equal to $2$,
[list=i]
[*] $A$ and $B$ have the same set of distinct prime factors.
[*] $A+1$ and $B+1$ have the same set of distinct prime factors.
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A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?
Decompose the polynomial
$$x^8 + x^4 +1$$
to factors of at most second degree.
Factor $a^{10} + a^5 + 1$ into nonconstant polynomials with integer coefficients