All Questions
Tagged with prime-factorization algebra-precalculus
28
questions
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:
$$
1000027=(100+3)(100^2-3\...
- 6,915
6
votes
2
answers
9k
views
Total number of divisors of factorial of a number
I came across a problem of how to calculate total number of divisors of factorial of a number. I know that total number of divisor of a number $n= p_1^a p_2^b p_3^c $ is $(a+1)*(b+1)*(c+1)$ where $a,...
- 298
4
votes
3
answers
113
views
Find the greatest common divisor of $2003^4 + 1$ and $2003^3 + 1$
Find the greatest common divisor of $2003^4 + 1$ and $2003^3 + 1$ without the use of a calculator. It is clear that $2003^4+1$ has a $082$ at the end of its number so $2003^4+1$ only has one factor of ...
- 321
3
votes
1
answer
55
views
A smart way to do this question.
Let $S=\{0,1,2,\dotsc,25\}$
And $T=\{n\in S : n^2+3n+2\text{ is divisible by }6\}$
Then the number of elements in $T$ is?
One way I know is to factorise it as $(n+1)(n+2)$.
And then put each $n$ ...
- 1,567
2
votes
3
answers
766
views
Find the prime factors of $3^{32}-2^{32}$
I'm having a go at BMO 2006/7 Q1 which states: "Find four prime numbers less than 100 which are factors of $3^{32}-2^{32}$."
My working is as follows (basically just follows difference of two squares ...
- 305
2
votes
2
answers
144
views
Finding positive integer $n>10$ that maximizes $\frac{\sigma_0(n)}{2^{\log n}}$
Among all the positive integer, which one integer, $n$, can make the number below the largest?
$$f(n)=\frac{\sigma_0(n)}{2^t}$$where $t=\log_{10}n$ and $\sigma_0$ is the divisor function.
For example,...
- 183
2
votes
2
answers
251
views
How does one prove that $(2\uparrow\uparrow16)+1$ is composite?
Just to be clear, close observation will show that this is not the Fermat numbers.
I was reading some things (link) when I came across the footnote on page 21, which states the following:
$$F_1=2+1\...
2
votes
1
answer
47
views
In $1 < k < n-10^6$, what is $k$? (details in question)
This is a homework question of mine, I am not searching for the solution but rather what it means. It seems pretty straight forward but I am a little confused as to what the $k$ in $1 < k < n-10^...
- 167
2
votes
1
answer
59
views
How to find the number of compound divisors of the smallest product from two unknown numbers?
The problem is as follows:
The number of panadol pills at a pharmacy is a positive whole number
that it has two prime divisors and 45 positive divisors. The number of
tylenol pills at the same ...
- 4,185
2
votes
1
answer
78
views
What are the other factors of x if we know 2, 4, and 9 are factors. [closed]
The factors of x include 2, 4, 9. Which of the following are also factors of x?
{1, 3, 5, 6, 8, 10, 12, 18, 24, 36}
Apparently the correct answer is {1, 3, 6, 12, 18, and 36} but I have trouble seeing ...
- 31
1
vote
3
answers
2k
views
Find the number of trailing zeros in 50! [duplicate]
My attempt:
50! = 50 * 49 *48 ....
Even * even = even number
Even * odd = even number
odd * odd = odd number
25 evens and 25 odds
Atleast 26 of the numbers will lead to an even ...
- 321
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:
$$\frac{b^2m-a\left(b+x\right)}{a+m\...
- 1,450
1
vote
2
answers
124
views
The Diophantine Equation $m(n-2016)=n^{2016}$
How many natural numbers, $n$, are there such that $$\frac{n^{2016}}{n-2016}$$ is a natural number?
HINT.-There are lots of solutions
HINT.-$\frac{n}{n-2016}=m \iff \frac{2016}{n-2016}=m-1$ and if, ...
- 30.3k
1
vote
1
answer
73
views
When $f(x) = \frac{ax + b}{a -x + 1}$ is an integer
Given that $a$ and $b$ are positive integers. and $$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