Tagged Questions
16
votes
1answer
135 views
To prove that $2^{3n}+2^n +1$ is not a perfect square.
Question: Prove that $2^{3n} + 2^n + 1$ cannot be a perfect square for any natural $n$.
I attempted this question and failed in two different ways.
1) I considered a polynomial $p(x) = x^3+ x + 1 - ...
5
votes
3answers
49 views
$p$ prime, $1 \le k \le p-2$ there exists $x \in \mathbb{Z} \ : \ x^k \neq 0,1 $ (mod p)
I found this problem in my algebra book, but unfortunately, there is no solution included. Here it is:
Let $p$ be a prime, $1 \le k \le p-2$. Show that there exists $x \in \mathbb{Z} \ $ such that ...
1
vote
4answers
69 views
I cannot find the last factor of this expression?
I'm supposed to factor $x^8-y^8$ (the exponents are 8 for both if it is too difficult to see) as completely as possible. It is easy to factor this to $(x+y)(x-y)(x^2+y^2)(x^4+y^4)$. However, the book ...
27
votes
6answers
552 views
Is $\sqrt[3]{p+q\sqrt{3}}+\sqrt[3]{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable?
In this recent answer to this question by Eesu, Vladimir
Reshetnikov proved that
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$
...
1
vote
0answers
68 views
Perfect power that has digits of $0$ and $6$ only in decimal notation. [closed]
Is there a positive integer that is a perfect power and has digits of $0$ and $6$ only in decimal notation?
30
votes
1answer
545 views
Proving that $x$ is an integer, if the differences between any two of $x^{1919}$, $x^{1960}$, and $x^{2100}$ are integers
For a specific real number $x$, the difference between any two of $x^{1919}$, $x^{1960}$ , and $x^{2100}$ is always an integer. How would one prove that $x$ is an integer?
3
votes
4answers
60 views
Solve : $\frac{n}{2}(n+1)=2014+2k$.
$n,k$ are positive integers and $n>k$, solve the equation : $$\frac{n(n+1)}{2}=2014+2k.$$
the first thing I did is to write the LHS as $(2n+1)^2$ but I face an equation like $ak+b=m^2$, I know ...
6
votes
1answer
59 views
Find the floor value of a finite continued surd
Given $x=20062007$, and let
$$A=\sqrt{x^2+\sqrt{4x^2+\sqrt{16x^2+\sqrt{100x^2+39x+\sqrt{3}}}}}.$$
Find the greatest integer not exceeding $A$.
0
votes
2answers
88 views
Find $C$ if $B(B-C)=23$. $B$ and $C$ are positive integers.
I don't know how to tackle the problem. I tried to factor the equation and use systems of equations but it still does not work. Please give a good proof.
2
votes
1answer
58 views
Conclusion about Zeros of a polynomial ,when sum of it's coefficients is zero
I have a polynomial of the form:
$$\sum_{m=0}^k\frac{(-1)^{m+1}(4k-2m)!x^{2k-2m}}{m!(2k-m)!(2k-2m+1)!}$$
or identically:
$$\sum_{m=0}^k\frac{(-1)^{m+1}(4k-2m)!(x^{2})^{k-m}}{m!(2k-m)!(2k-2m+1)!}$$
...
5
votes
2answers
96 views
How many cups of sugar do I need for these 5th grade problems?
Problem 1a: If 4 glasses of a mixture needs 1 cup of sugar how many cups of sugar are needed for 5 glasses?
This one is easy and makes sense. It's just simply $\frac{1}{4}*5$ Now taking it a notch ...
0
votes
2answers
91 views
Root of a polynomial with rational coefficients
I am currently learning about Direct Proofs. I am struggling trying to find a starting point to prove the Statement: For all real numbers $c$, if $c$ is a root of a polynomial with rational ...
3
votes
1answer
100 views
Property true for some integers and false for others: $-a^n$ = $(-a)^n$
I am currently working in my Discrete math class with elementary number theory and methods of proof. I have been given the problem $-a^n = (-a)^n$. According to the professor and the book this ...
4
votes
1answer
81 views
Can the distance from the vertices of a square of integer width to an inscribed circle all be integer?
I'm looking for solutions to the following British Mathematical Olympiad question:
Suppose that $ABCD$ is a square and that $P$ is a point which is on the circle inscribed in the square. Determine ...
4
votes
1answer
61 views
How can the formula be found for this problem?
We have a truck that we need to completely fill up with merchandise. We have an infinite supply of merchandise of dimension $1\times1\times1, 2\times2\times2, 4\times4\times4, 8\times8\times8, ...
0
votes
2answers
79 views
Find all $q\in\mathbb{Q}$ such that $ qx^2+(q+1)x+q=1 $ has integer solutions.
Find all $q\in\mathbb{Q}$ such that equation $$ qx^2+(q+1)x+q=1 $$ has integers as solutions.
I tried solving it for $x$ ($q\ne0$) and stating $\sqrt{D}=\sqrt{-3q^2+6q+1}$ has to be rational, so ...
2
votes
3answers
115 views
Something kind of like proving the euclidean Algorithm by induction
Let a > b be positive integers. In applying the Euclidean algorithm,
we have $a = b q_0$ + $r_0$, $b = r_0 q_1 + r_1$, and $r_{n-1} = r_n q_{n+1} + r_{n+1}$, for all $n > 0$. Prove by induction ...
23
votes
0answers
804 views
$4494410$ and friends
$4494410$ has the property that when converted to base $16$ it is $44944A_{16}$, then if the $A$ is expanded to $10$ in the string we get back the original number. ...
4
votes
4answers
265 views
Next number after 1729??
I have known this from beginning that
1729 is the smallest number expressible as the sum of two cubes in two different ways:
$$ 12^3 + 1^3 $$
and
$$ 10^3+9^3 $$
I am a Software Developer and ...
3
votes
0answers
59 views
Polynomial bound
Let $P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$ such that
$$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$
Suppose that $P(x)> 0$ for all ...
0
votes
1answer
49 views
Proof of floor of division
I got stuck proving
$$\left\lfloor\frac{x/a}b\right\rfloor = \left\lfloor\frac{\lfloor x/a\rfloor}b\right \rfloor$$
This is what I got:
Using the division algorithm we can write $x = qa+r$, where ...
0
votes
2answers
42 views
Fit screen resolution given ratio and total number of pixels
Given:
width: 1920
height: 1080
total pixels: width * height = 2073600
aspect ratio: 1920 / 1080 ~= 1.8
How do I calculate a new resolution (width and height) ...
2
votes
1answer
105 views
Establishing an inequality using principal convergents and continued fraction representation.
If $\theta$ is irrational with continued fraction representation $[0;a_1,a_2,\ldots]$, $\lbrace \frac{m_k}{n_k} \rbrace$ is the sequence of principal convergents of $\theta$ and $\lbrace b_k\rbrace$ ...
1
vote
4answers
87 views
Number of ordered sets of integers
How many ordered sets of integers $(x,y,z)$ satisfying $$x,y,z \in [-10,10]$$ are solutions to the following system of equations:
$$x^2y^2+y^2z^2=5xyz$$
...
8
votes
1answer
108 views
Why cannot $2^x=m^n+1$?
$x,n,m >1$ and $x,n,m \in \mathbb{Z}$
I've tried to solve it myself, but I'm getting nowhere, so apologies if it's an irritatingly basic question.
Whilst I'm on it, is it true that $y^x \ne ...
1
vote
2answers
129 views
Number of solutions for $\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}$ where $1 \leq N \leq 10^6$
Note: this is a programming challenge at this site
For this equation $$\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}\quad ( N \text{ factorial} ),$$
find the number of positive integral solutions for ...
6
votes
3answers
345 views
The positive integer solutions for $2^a+3^b=5^c$
What are the positive integer solutions to the equation
$$2^a + 3^b = 5^c$$
Of course $(1,\space 1, \space 1)$ is a solution.
8
votes
3answers
166 views
$a+b=c \times d$ and $a\times b = c + d$
There is a 'nice' relationship between the integers (1,5) and (2,3) as
$$1+5=2 \times 3;$$
$$1\times 5 = 2 + 3.$$
So I tried to find all positive integers pairs $(a, b)$ and $(c, d)$ such that ...
0
votes
1answer
61 views
Why don't all elements of an arithmetic progression divide the lcm of the start and step?
I'm getting back into basic proofs after a long hiatus, and I know something has to be wrong with the following logic but I'm not sure what.
Elements of arithmetic progressions can be expressed as:
...
0
votes
1answer
65 views
amortized analysis
a) define f(k) as the largest power of 2 that divides k.
For example, f(25) = 1, f(42) = 2, f(144) = 16.
What is ${1 \over k}\sum_1^k f(k)$?
b) define f(k) as the square of largest power of 2 that ...
1
vote
2answers
103 views
number of integral solutions for $x^2+y^2=5^k$
Prove that the equation $x^2+y^2=5^k$ has $4k+4$ integral solution.
Any ideas would be appreciated.
Thanks
1
vote
3answers
286 views
Are there any integer solutions to $a^3=b^2$?
I was wondering if there were any two integers $a$ and $b$ where $a^3=b^2$.
1
vote
1answer
106 views
For integers $a$ and $b \gt 0$, and $n^2$ a sum of two square integers, does this strategy find the largest integer $x | x^2 \lt n^2(a^2 + b^2)$?
Here is some background information on the problem I am trying to solve. I start with the following equation:
$n^2(a^2 + b^2) = x^2 + y^2$, where $n, a, b, x, y \in \mathbb Z$, and $a \ge b \gt 0$, ...
3
votes
2answers
79 views
Question regarding what appears to be an identity
This is an MCQ we were posed in school recently (I hope you don't mind elementary stuff):
What is $(x-a)(x-b)(x-c)...(x-z)$ ?
Options:
$0$
$1$
$2$
$(x^n)-(abcdef...z)$
1
vote
2answers
66 views
A Trivial Question - Increasing by doubling a number when its negative
The question is : if $x=y-\frac{50}{y}$ , where x and y are both > $0$ . If the value of y is doubled in the equation above the value of x will a)Decrease b)remain same c)increase four fold ...
4
votes
1answer
117 views
If $A,B$ are factors of $2^6 3^4 5^2,$ how many values of $|A-B|$ are possible?
Let $x=2^6 3^4 5^2$, then how many distinct values of $|A-B|$ are possible where $A, B$ are the factors of $x$?
How to approach this problem?
1
vote
1answer
66 views
Solving Congruences and CRT
I have never really directly dealt with congruences until I was introduced to the Chinese Remainder Theorem. Although there are tons of different versions of this theorem out there, currently I am ...
3
votes
4answers
129 views
Finding the divisor of an unknown
I am trying to solve this problem
A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the ...
3
votes
4answers
845 views
Finding a number when its remainder is given.
I am trying to solve this problem
W is a positive integer when divided by 5 gives remainder 1 and when divided by 7 gives remainder 5. Find W.
I am referring back to an earlier post I ...
0
votes
2answers
166 views
Solving a fraction expression without a calculator using properties
I am preparing for the CAT and am not allowed to use a calculator to solve questions like these
If k is an integer and $\frac{35^2-1}{k}$ is also an integer then k could be any of the ...
0
votes
3answers
373 views
Calculating a number when its remainder is given
I am having difficulty solving the following problem:
Marge has n candies , where n is an integer between 20 and 50.If marge divides the candies equally among 5 children she will have 2 ...
0
votes
3answers
112 views
Remainder problem
If the remainder when $x$ is divided by 5 equals to the remainder when x is divided by 4 then x could be any of the following a)20 , b)21 , c)22 , d)23 e)24 (Ans=(e)24)
Now I could only ...
1
vote
3answers
83 views
Checking divisibility of an expression - Need Pointers
I would like it if someone could give me pointers on solving problems like these. And why was 4 the answer here ?
If $a=4b+26$ and $b$ is positive , then a could be divisible by all of ...
2
votes
2answers
63 views
Finding the remainder from equations.
I am having problems solving this question :
When n is divide by 4 the remainder is 2 what will the remainder be when 6n is divided by 4 ? Ans=$0$
Here is what I have got so far
...
5
votes
4answers
308 views
If $n = m^3 - m$ for some integer $m$, then $n$ is a multiple of $6$
I am trying to teach myself mathematics (I have no access to a teacher), but I am not getting very far. I am just working through the exercises at the end of the book's chapter, but unfortunately ...
0
votes
2answers
69 views
Can't understand this remainder solution
The problem is:
$W$ is a positive integer when divided by $5$ gives remainder $1$ and when divided by $7$ gives remainder $5$. Find $W$.
Answer: Take the larger divisor , So Expression ...
1
vote
4answers
82 views
Divisibility proof involving substitution
I’m having trouble with the proving the following:
If $a \mid b-2c$ and and $a \mid 2b+3c$ then $a \mid b$ and $a \mid c$.
Heres my partial solution:
By definition of divisibility, $b-2c=ak$ ...
0
votes
0answers
64 views
Calculating A from this equation
I am having trouble with the following question
If A and B are positive integers and $A^2 + B^2 = 36$ Then what is $A$? The choices are 6, 7, 8, 9, or 10.
How does one show that answer ...
3
votes
2answers
264 views
Some digit summation problems
What is the sum of the digits of all numbers from 1 to 1000000?
In general, what is the sum of all digits between 1 and N?
f(n) is a function counting all the ones that show up in 1, 2, 3, ...
0
votes
1answer
163 views
Simultaneous equations
My question is: Solve simultaneously:(anwers are in integers)
$$\begin{align} y^3 - 9x^2 + 27x - 27 &= 0 \\
z^3-9y^2+27y-27 &= 0 \\
x^3-9z^2+27z-27 &= 0 \end{align}$$
Any hints to solve ...
