Is this even possible to solve?  30 is an even number. I don't think there's Answer for this .
107093
I've been asked the following question:  What three odd integers from the set {1,3,5,7,9,11,13,15} that when summed together equals to 30? Note that any integer can be used more than once.  Is there any possibility to solve these kind of questions with some formulae? Note I have gone through the answers for making it with 5 numbers.
58186
Help me understand problems of this type a bit more intuitively.   The solution $C(3+11−1,11)$ seems simple enough, but I got stuck thinking about how many integers you are choosing from within $x_1$, $x_2$, and $x_3$ and I was trying to get an $n$ - as in $C(n+r-1,r)$ - this way. Also, I though the $r$ was 3 since you have $x_1$, $x_2$, $x_3$ choices. Basically, I went wrong at every step. Help!
56856
First of all, $n$ is even (find that out by trying to see what kind of remainders you can possibly get when dividing everything by 4). So $n=2m$, $m\in N$.  $ x^2 + 8 = 3^{2m} \Rightarrow x^2 - 1 = 3^{2m} - 9 \Rightarrow (x+1)(x-1) = (3^m + 3)(3^m - 3) \Rightarrow x$ is odd.  That's all I've tried so far. I'd like to get some help. Thanks.
21724
A cube has numbers ±1 placed at each vertex and each center of cube’s six faces (so 14 numbers altogether). Moreover, it is known that each number in the center of a square face equals to the product of the 4 numbers at the vertices of this face. Can the sum of all 14 numbers be equal to zero? My first thoughts were test out some combinations and look for a pattern, but I was not able to notice anything major about it. So, the only progress I was able to really make was breaking it down into $V_1*V_2*V_3*V_4$, and so on for each face. Any help is greatly appreciated.
30883
If $a$ is an odd integer, prove that $24 \mid a(a^2-1)$.  I proved in my class that $a^2=8k+1$ for some $k$, provided $a$ was odd.  So $$a(a^2-1)=(2k-1)(8k+1-1)=8k(2k-1)$$ but I can't pull a $24$ out of it.
67560
I am stuck on the 4 th one.I have done the rest.I have found out the value of a_n.But not getting how to proceed for the 4 th one.
9540
Fix integer $n\geq 1$. Consider the number $$1-\binom{n}{2}+\binom{n}{4}-\binom{n}{6}+\cdots$$where the sum continues as long as the lower number in the binomial is $\leq n$. Is there a way to simplify this sum?   The first few values are $1, 0, -2, -4, -4$.
41621
Is it possible for some integer $n&gt;1$ that $2^n-1\mid 3^n-1$ ?   I have tried many things, but nothing worked.
72327
How we can show every permutation is either even or odd,but not both......I can't arrive at a proof for this ..... Can anybody give me the proof...  Thanks in advance...
90615
I have this problem: How many integers between $1$ and $10^4$ contain exactly one $8$ and one $9$?, but i dont know if $18, 19, 28, 29...$ serve, else it could be $8$ y $9 = 2$ numbers, help please.
41147
Problem :   Solve in $\mathbb N$ :   $8n+9\equiv 0\pmod{1163}$  The mathematics give me : $n=435+1163k$ , $k\in\mathbb N$   $1163$ is a prime number   $8n\equiv -9\pmod{1163}$   $8n\equiv 1154\pmod{1163}$   I don't know any ideas to complete my work ?   I have to see your hints
122043
What I get is $4 \cdot 54k + 27(n(n-1)) - 2$.  I understand first two terms but $-2$ is giving me a headache. I suppose to be true every term should be divisible by $54$. Or did I make wrong conclusion?  P. S: Is there mathjax tutorial?
33216
I considered an even number $n\geq 10$, where it is divisible by some positive integer $k$.  Also, $k$ divides $\frac{n}{2}$. Then $n = kq\implies k\cdot\frac{q}{2} = \frac{n}{2}$.  Can we say that $q$ is even in this case? Can anyone help to get it theoretically? Thanks a lot for the help.   PS.   From the answer given by Bill, it does not satisfy if we take $n=18,k=3,j=9.$ Kindly help. Let me rectify where I am going wrong. Please help.  $\quad\ \ \ \ \ \ \ \ \ $ If $\,\rm\ J,K\mid N\,\ $  then $\rm\,\ \color{#c00}K\ {\LARGE \mid} {\Large  \frac{N}{\color{#0a0}J}}\! \iff\color{#0a0}J\ {\LARGE \mid} {\Large \frac{N}{\color{#c00}K}}\, $ [$\rm\!\!\iff\!\!  JK\mid N\,$]  This part is not satisfied.
58471
How many $4$ digit even numbers have all $4$ digits distinct ?   This question has been asked if we replace even with positive.   I was solving this Problem like: For first position $1-9$ (No $0$) for next $0-9$ (leaving out the digit which has been choose) for next $0-9$ (leaving out $2$ digits), but on last digit I get stuck. From what I should choose. Will it be $0, 2, 4, 6, 8$, because number needs to be even? How should I solve it?
38763
I was just getting my hands dirty solving some equations of the form $x=\phi(n)$ where $\phi(n)$ is Euler totient function. I know that $\phi(n)$ is even for $n\geq 3$. However, I am wondering that:     Can all even integers $x$ be expressed as $\phi(n)$ where $n$ is an integer?   I start doing some research, the first even number appears to hove no solution is $14$. According to , there is no solution to the equation $14=\phi(n)$ for $n\leq 500$. But does it have a solution for $n&gt;500$?  Moreover, what can you say about the above statement?
100491
Find the smallest integer $n$ such that $\frac{11}{15}&gt;\frac{m}{n}&gt;\frac{7}{10}$ and $m$ is an integer. There are multiple ways to solve this problem. We can get rid of the fractions by multiplying by $30n$ and do casework, we can convert to decimals and do casework, heck, we can graph and do casework. I know that this problem is really simple, but I am just looking for a way to solve this problem without casework.
11316
How can I solve $x^2 \equiv 19 \pmod {59}$?  I know that we can just try squaring numbers from 1 to 58 , but this is a very slow method, is not their a quicker one?