Found problems: 788
Two three-digit numbers are given. The hundreds digit of each of them is equal to the units digit of the other. Find these numbers if their difference is $297$ and the sum of digits of the smaller number is $23$.
Let $s (n)$ denote the sum of digits of a positive integer $n$. Using six different digits, we formed three 2-digits $p, q, r$ such that $$p \cdot q \cdot s(r) = p\cdot s(q) \cdot r = s (p) \cdot q \cdot r.$$ Find all such numbers $p, q, r$.
Prove that the first 3 digits after the decimal point in the decimal expression of the number $\frac{0.123456789101112 . . . 495051}{0.515049 . . . 121110987654321}$ are $239$.
Determine all three-digit numbers N such that $N^2$ has six digits and so that the sum of the number formed by the first three digits of $N^2$ and the number formed by the latter three digits of $N^2$ equals $N$.
Which positive integers satisfy the following three conditions?
a) The number consists of at least two digits.
b) The last digit is not zero.
c) Inserting a zero between the last two digits yields a number divisible by the original number.
Does there exist a quadratic trinomial $p(x)$ with integer coefficients such that, for every natural number $n$ whose decimal representation consists of digits $1$, $p(n)$ also consists only of digits $1$?
How many three-digit positive integers have an odd number of even digits?
$\textbf{(A) }150\qquad\textbf{(B) }250\qquad\textbf{(C) }350\qquad\textbf{(D) }450\qquad\textbf{(E) }550$
Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.
Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.
A positive integer $N$ is called [i]northern[/i] if for each digit $d > 0$, there exists a divisor of $N$ whose last digit is $d$. How many [i]northern [/i] numbers less than $2016$ are there with the fewest number of divisors as possible?
Exists a positive integer $n$ such that the number $\underbrace{1...1}_{n \,ones} 2 \underbrace{1...1}_{n \, ones}$ is a prime number?
For a fixed $k$ with $4 \le k \le 9$ consider the set of all positive integers with $k$ decimal digits such that each of the digits from $1$ to $k$ occurs exactly once.
Show that it is possible to partition this set into two disjoint subsets such that the sum of the cubes of the numbers in the first set is equal to the sum of the cubes in the second set.
We call the [i]tribi [/i] of a positive integer $k$ (denoted $T(k)$) the number of all pairs $11$ in the binary representation of $k$. e.g $$T(1)=T(2)=0,\, T(3)=1, \,T(4)=T(5)=0,\,T(6)=1,\,T(7)=2.$$
Calculate $S_n=\sum_{k=1}^{2^n}T(K)$.
Consider the integers formed using the digits $0,1,2,3,4,5,6$, without repetition. Find the largest multiple of $55$. Justify your answer.
Find all two-digit numbers $x$ the sum of whose digits is the same as that of $2x$, $3x$, ... , $9x$.
Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.
Let us denote by $S(m)$ the sum of the digits of the natural number $m$. Prove that there are infinitely many positive integers $n$ such that $$S(3^n) \ge S(3^{n+1}).$$
Find all natural numbers $n$ with the following property:
Given the decimal writing of $n$, adding a few digits one can obtain the decimal writing of $1996n$.
A natural number of five digits is called [i]Ecuadorian [/i]if it satisfies the following conditions:
$\bullet$ All its digits are different.
$\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$, but $54210$ is not since $5 \ne 4 + 2 + 1 + 0$.
Find how many Ecuadorian numbers exist.
Determine the smallest integer of the form $\frac{ \overline{AB}}{B}$ .where $A$ and $B$ are three-digit positive integers and $\overline{AB}$ denotes the six-digit number that is form by writing the numbers $A$ and $B$ consecutively.
Prove that every real positive number may be represented as a sum of nine numbers whose decimal representation consists of the digits $0$ and $7$.
(E Turkevich)
For every positive integer $n$, $s(n)$ denotes the sum of the digits in the decimal representation of $n$. Prove that for every integer $n \ge 5$, we have $$S(1)S(3)...S(2n-1) \ge S(2)S(4)...S(2n)$$
Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.
Let $s(n)$ denote the sum of digits of a positive integer $n$. Prove that $s(9^n) > 9$ for all $n\geq 3$.
We have a positive integer $n$, whose sum of digits is $100$ . If the sum of digits of $44n$ is $800$ then what is the sum of digits of $3n$?