All Questions
10
questions
1
vote
1
answer
165
views
If $PQRSPQRS$ is an eight-digit number with $56$ divisors, find the number of divisors of $PQRS$.
$PQRS$ is a four-digit number where $P,Q,R,S$ are the digits of the number. If $PQRSPQRS$ is an eight-digit number with $56$ divisors (including $1$ and $PQRSPQRS$), find the number of divisors of $...
- 8,338
1
vote
2
answers
287
views
Relatively prime factors of $24500$
Let $N=24500$, then find the number of ways by which $N$ can be resolved into two coprime factors?
My tries:
$N=24500=2^2\cdot 5^3\cdot 7^2$, for co prime no those two factors of $24500$ should ...
- 1,943
1
vote
4
answers
98
views
When $f(x)=\frac{-b^2m-ba+ax}{-mx-bm-a}$ is an integer
$a,b,m,x$ are positive integers.
For which $x>0$ is $f(x)$ an integer?
$$f(x)=\frac{-b^2m-ba+ax}{-mx-bm-a}$$
I been trying to play with it, I changed it to:
$$\frac{b^2m-a\left(b+x\right)}{a+m\...
- 1,450
1
vote
1
answer
73
views
When $f(x) = \frac{ax + b}{a -x + 1}$ is an integer
Given that $a$ and $b$ are positive integers. and $$f(x) = \frac{ax + b}{a -x + 1}$$ is an integer, what integer values can x have?
If I could only somehow move $x$ from numerator to the denominator ...
- 1,450
0
votes
2
answers
427
views
What I'm missing in $p=3^{q}\cdot 2^{r}$?
Given positive integer $p,q,r$ with $p=3^{q}\cdot 2^{r}$ and $100<p<1000$. The difference between maximum and minimum values of $(q+r)$ is?
My tries
It's clear that $p=6k$ for some positive ...
- 1,943
10
votes
5
answers
796
views
Find all the prime factors of $1000027$
Find all the prime factors of $1000027$.
I got all the factors by testing every number from $1$ to $103$, but when I try to do it using algebra, I get stuck.
My work:
$$
1000027=(100+3)(100^2-3\...
- 6,915
1
vote
2
answers
124
views
The Diophantine Equation $m(n-2016)=n^{2016}$
How many natural numbers, $n$, are there such that $$\frac{n^{2016}}{n-2016}$$ is a natural number?
HINT.-There are lots of solutions
HINT.-$\frac{n}{n-2016}=m \iff \frac{2016}{n-2016}=m-1$ and if, ...
- 30.3k
6
votes
2
answers
9k
views
Total number of divisors of factorial of a number
I came across a problem of how to calculate total number of divisors of factorial of a number. I know that total number of divisor of a number $n= p_1^a p_2^b p_3^c $ is $(a+1)*(b+1)*(c+1)$ where $a,...
- 298
4
votes
3
answers
113
views
Find the greatest common divisor of $2003^4 + 1$ and $2003^3 + 1$
Find the greatest common divisor of $2003^4 + 1$ and $2003^3 + 1$ without the use of a calculator. It is clear that $2003^4+1$ has a $082$ at the end of its number so $2003^4+1$ only has one factor of ...
- 321
0
votes
0
answers
382
views
Positive integers of sum and products
Find all pairs of positive integers $m$ and $n$ where $m<n$ such that the sum of $m$ and $n$ added to the product of $m$ and $n$ is equal to $2014$
I just thought about this question and wanted to ...
- 167