3
votes
3answers
84 views

How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for ...
1
vote
1answer
47 views

Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which says what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
0
votes
2answers
49 views

how shall i find the $n$-th term of this,

How shall I find the $n$-th term of this: $\sqrt{1+2}$ $\sqrt[3]{1+2+3}$ $\sqrt[4]{1+2+3+4}$ $\sqrt[5]{1+2+3+4+5}$ $\sqrt[6]{1+2+3+4+5+6}$ $\sqrt[7]{1+2+3+4+5+6+7}$ all the way to ...
4
votes
1answer
47 views

Find integral solutions for $2x^2+y^2=2\times(1007)^2+1$

Find integral solutions to the equation $$2x^2+y^2=2\times(1007)^2+1$$ I tried: I rewrote the equation as $2x^2+y^2=2028099$. I found that $y_{max}=1424$ and $y$ must be odd, so I set ...
0
votes
1answer
77 views

Is this real number an integer?

Is this real number : $$\Big(2+\frac{10}{9}\sqrt{3}\Big)^{1/3}+\Big(2-\frac{10}{9}\sqrt{3}\Big)^{1/3}$$ an integer ? I've tried different factorization, but nothing seems to work.
2
votes
3answers
110 views

Mathematical way to solve integer numbers $217 = (20x+3)r+x$

Is there any mathematical way to find the integer numbers that solve the following equation: $$217 = (20x+3)r+x$$
1
vote
2answers
53 views

How to simplify the formula for $n$th Fibonacci number when $n=2$?

When n is equal to 2 how do I simplify when the $n=2$ is put into the equation below (by the way I have to prove this formula by induction that when n= any number it will equal that number) ...
2
votes
3answers
83 views

searching smallest number that has $40$ distinct positive divisors

What is the smallest natural number such that it has $ 40 $ distinct positive (integer) divisors (inclusive of $ 1 $ and itself? At first I was stunned of seeing the problem.It's not possible to find ...
2
votes
1answer
47 views

Given $n$, find smallest number $m$ such that $m!$ ends with $n$ zeros

I got this question as a programming exercise. I first thought it was rather trivial, and that $m = 5n$ because the number of trailing zeroes are given by the number of factors of 5 in $m!$ (and ...
3
votes
2answers
55 views

Prove that every non-prime natural number $ > 1$ can be written in the form of $n+(n+2)+(n+4)+…+(n+2m) = p$

I'm trying to prove that every non-prime natural number greater than $1$ can is equal to a sum of consecutive even or odd numbers. This can be resumed in : « $p,m,n \in ℕ$» , «$p > 1$» , «$n > ...
2
votes
2answers
127 views

Counting the number of digits in a concatenation

Concatenate the numbers $2^{1971}$ and $5^{1971}$. How many digits are there in the new number? How do I count them?
3
votes
1answer
112 views

Remainder of $\frac{x^{60}+x^{48}+x^{36}+x^{24}+x^{12}+1}{x^{5}+x^{4}+x^{3}+x^{2}+x+1}$

I am trying to find the remainder of the polynomial division $$\frac{x^{60}+x^{48}+x^{36}+x^{24}+x^{12}+1}{x^{5}+x^{4}+x^{3}+x^{2}+x+1}$$ I know that the answer is 6, but I am not getting that when I ...
2
votes
0answers
113 views

Evaluate this product $n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^k-1)}{2^k}$

For $k = \lfloor \log_{2}(n+1) \rfloor - 1$ evaluate $n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{k}-1)}{2^k}$ So the product goes up to $k$ and I ...
0
votes
0answers
21 views

How to find a certain uppper bound (see details)?

What would be the most efficient way to find this upper bound? Given natural number n and a natural number d < n, find the ...
0
votes
1answer
43 views

How to derive this formula about the bracket function?

Is there a direct way of proving that $$ [nx] = [x] + [x+\frac{1}{2}] + [x+\frac{1}{3}] + \ldots + [x+ \frac{1}{n}]$$ for each real number $x$ and for each positive integer $n$? My effort: Let ...
1
vote
3answers
63 views

$3$ doesn't divide $x\Longrightarrow\;x^3\equiv\pm1 (\operatorname{mod}9)$ [closed]

I'm stuck in this elementary problem: how can I show that $3$ doesn't divide $x$ implies $\;x^3\equiv\pm1 (\operatorname{mod}9)\:$? Thanks a lot
0
votes
2answers
79 views

Simple math pattern--does it work?

So a friend of mine just pointed this out: $$ \text {If} \ \; 0<a<b \; \text{then} $$ $$ b^3-a^3=(a^2+ab+b^2)(b-a) $$ $$ b^4-a^4=((a^3)+(a^2b)+(ab^2)+(b^3))(b-a) $$ $$ ...
14
votes
2answers
230 views

Integer values of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$?

What are the possible integer values of $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$$ where $x$, $y$, and $z$ are positive integers? My suspicion is the the only integer values are $3$ and $5$, the former ...
3
votes
1answer
57 views

For what positive integers $p$ and $q$: $(p+1)!+(q+1)!=(pq)^2$

I tried this problem using brute force and got the answers as $(3,4)$ and $(4,3)$,but is there a way to solve this question?
5
votes
1answer
119 views

Prove that $512^3 + 675^3 + 720^3$ is a composite number

We have to prove that the number $$N=512^3 + 675^3 + 720^3$$ is composite. I tried to use the identity $(a^3+b^3+c^3)=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)+3abc$ hoping to take out some common ...
0
votes
1answer
44 views

prove that $rp+1$ does not divide $p^p-1$

$p$ is a odd prime and $r$ is any odd positive integer. prove that $rp+1$ does not divide $p^p-1$ for any $p$ & $r$. I have expanded $p^p-1$ to $p^p-1=(p-1)(p^{p-1}+p^{p-2}+...+1)$ since ...
4
votes
1answer
45 views

Prove that $\lceil \frac{\sqrt{n^2+1+\sqrt{n^2}}}{\sqrt{n^2+3+\sqrt{n^2+2}}-\sqrt{n^2+2+\sqrt{n^2+1}}}\rceil = 2n^2+n+3$

First, the question: Prove that $$\Bigg\lceil \frac{\sqrt{n^2+1+\sqrt{n^2}}}{\sqrt{n^2+3+\sqrt{n^2+2}}-\sqrt{n^2+2+\sqrt{n^2+1}}}\Bigg\rceil = 2n^2+n+3$$ The motive to this question is the ...
1
vote
1answer
147 views

Is $\pi = 4$ really? [duplicate]

Can anyone explain what's wrong with this?
12
votes
3answers
409 views

Number Theory or Algebra?

Prove that if $4^m-2^m+1$ is a prime number, then all the prime divisors of $m$ are smaller than $5$ I initially thought about putting $4^m-2^m+1=p$ where $p$ is some prime and after eliminating ...
6
votes
4answers
156 views

Prove that there is an integer $n$ such that $n^{1992}$ starts with $1992$ one's.

This was taken from an old Brazilian Mathematical Olympiad (1992). As the title says, we're supposed to prove that there is an integer $n$ such that $n^{1992}$ starts with $1992$ one's (in the ...
0
votes
3answers
73 views

Find primes $p_1,p_2,..,p_6$ such that $1+\prod_{i=1}^{6}p_i $is not prime

Show that if$$ p_1, p_2, p_3, p_4, p_5, p_6 $$are primes, then $$1+\prod_{i=1}^{6}p_i$$ is not necessarily prime by using a specic example.
1
vote
3answers
86 views

Prove: for all integers $x,y\in x+y+xy = 0$ -> $x=y=0 $ or $ x=y=-2$

How would you do the proof for this problem? Prove that for all integers $x,y$, $(x+y+xy=0)\implies ((x=y=0)\vee(x=y=-2))$
0
votes
0answers
35 views

The elegant expression in terms of gcd and lcm - algebra - (2)

Definition: suppose a quantity $P$ is identified by $$ \frac{P}{k}\simeq \frac{P}{k}+1 $$ what we mean is that $$ P= 0\pmod{k}. $$ That means that when $P \to P+k$, then $$ \frac{P}{k}\to ...
2
votes
1answer
42 views

The elegant expression in terms of gcd and lcm - algebra

Given three positive integer numbers $k_1$, $k_2$, $k_3$, we may denote their greatest common divisor(gcd) by $\gcd(k_i,k_j)\equiv k_{ij}$ for gcd of a two pair of number $k_i,k_j$. ...
2
votes
2answers
95 views

Why can't absolute values be expressed with negative numbers. [closed]

The answer to this question seems obvious. 'An absolute value expresses the quantity of ones between any number and 0'. But does that mean it must be positive? I took a shot at answering my ...
0
votes
2answers
83 views

How to solve the following? $\left\lfloor\dfrac x5 \right\rfloor- \left\lfloor\dfrac x7 \right\rfloor=1$ [on hold]

How do we find all positive integers $x$ such that $$\left\lfloor\dfrac x5 \right\rfloor- \left\lfloor\dfrac x7 \right\rfloor=1.$$
2
votes
2answers
41 views

How do we know $ n $ is a multiple of $ 2 $ from the equation $ 2 =\frac{ n^2} { d^2} $?

My attempt at answering starts by observing that if a number $ n $ is a multiple of $ 2 $, then it can be written in the form $ n = i \cdot 2 $ where $ i $ is some integer. Now I assume that there is ...
2
votes
2answers
88 views

If $x_1^3+x_2^3+\ldots+x_t^3=2002^{2002}$, find minimum value of $t$ so the condition can be satisfied by some natural numbers $x_i$

If $x_1^3+x_2^3+\ldots+x_t^3=2002^{2002}$, find the minimum value of $t$ so the condition can be satisfied by some natural numbers $x_i$. My attempt: I took modulo $9$ on both sides and found the ...
2
votes
4answers
80 views

Contest problem involving primes and factorization

Prove that for any nonnegative integer $n$, the number $$5^{5^{n+1}}+ 5^{5^{n}}+1$$ is not prime. I want only some hints and the method to follow, but I don't need the full solution. Thanks.
7
votes
2answers
240 views

Writing square root of square-free numbers as sum of square roots.

Some days ago i came across a question about writing $\sqrt {2001}$ as sum of two other square roots. I managed to prove that this is not possible unless one of them is zero and the other one is ...
0
votes
3answers
35 views

How to show that if $24k+4$ and $24k+1$ are both perfect squares where $k$ is a natural number, then it is only when $k=0$?

I am trying to show that if $24k+4$ and $24k+1$ are both perfect squares where $k$ is a natural number, then the only possibility is when $k=0$. Here is how I did it: Let $24k+4=x^2$ and $24k+1=y^2$, ...
1
vote
0answers
50 views

$x_1,x_2,\ldots,x_m$ is a permutation of $1,2,\ldots,m$ and $n_1,n_2,\ldots,n_m$ be integer and $m>1$ and odd.

$x_1,x_2,\ldots,x_m$ is a permutation of $1,2,\ldots,m$ and $n_1,n_2,\ldots,n_m$ be integer and $m>1$ and $m$ is odd. Now, $f(x)$ be defined as $f(x)=\sum\limits_{i,j=1}^{m} n_ix_j$. If $a,b$ are ...
0
votes
2answers
50 views

trying to teach negative times negative number positive answer

I am thinking of negative times negative numbers as: $$-3 \cdot 2 = (-2) + (-2) + (-2) = -6$$ $$2 \cdot -3 = (-3) + (-3) = -6$$ $$-3 \cdot -2 = (-(-2)) + (-(-2)) + (-(-2)) = 6$$ I am trying ...
3
votes
1answer
45 views

Finding all possible values

we have to find all possible prime values $(p,q,r)$ such that $ pq = r + 1 $ $ 2(p^2+q^2) = r^2 + 1 $ I do not know how to start looking for an answer.
3
votes
1answer
59 views

On $\lfloor\sqrt n \rfloor+ \sum_{j=1}^n \lfloor n/j\rfloor$

How do we prove that $\Big[\sqrt n \Big]+ \sum_{j=1}^n \bigg[ \dfrac nj\bigg]$ is an even integer for all $ n \in \mathbb N$ ? (where $\Big[ \space \Big]$ denotes the "greatest integer" function)
0
votes
2answers
75 views

Is there a formula for a simple sum of sums?

Given $f(n) = 1 + (1 + 2) + (1 + 2 + 3)+ \cdots + (1 + 2 + 3 +\cdots + n)$, I am wondering if there is a straightforward formula to compute f(n) and how it may be ...
8
votes
2answers
320 views

why is $\sqrt[3]{31}$ so close to $\pi$?

$\sqrt[3]{31}$ is about $3.14138$. Why is this so close to pi?
2
votes
1answer
77 views

The smallest 12 digit natural number for which the sum of its digits is 80.

So everytime I heard the word smallest natural number of 5 digits for example I do this in my head: 100000... ^ | which means that there must be a one in there: ...
1
vote
1answer
72 views

How to prove no integer solutions?

What I want to do is prove that there are no integers x, y that satisfy this equation: $11=x^2-3y^2$ I tried solving and other methods, but to no avail. Any help would be awesome.
0
votes
2answers
36 views

Prove that this equation have an non finite number of prime solutions

So the question seeks to answer the following, let $x,y\in\Bbb R$. Prove that there is a non finite number of prime solutions to the following equation: $3x-5y=11$. Our professor says that it's easy ...
1
vote
4answers
82 views

Pre College Mathematics

During my school days I was a very keen student of mathematics. But circumstances led me to opt for commerce at the college level. Now I wish to continue learning mathematics on a self study basis. ...
1
vote
1answer
93 views

Does an expression exist such that…

Can you prove or disprove the existence of an expression P, such that $Z=6ab+a+b-P$ Makes Z expressible in the form; $Z=6xy\pm x \pm y$ for all a and b where $a,b,x,y∈N $ Finding an example of ...
1
vote
2answers
151 views

How to prove that certain integers and xy are solutions for a relation?

I am trying to solve the following problem: Let A be the set of all integers of the form a^2 + b^2 + 4ab where a and b are integers. Prove: a. if x and y are in A, prove xy is in A. b. Prove 121 is ...
0
votes
1answer
63 views

Proving $\left\lfloor n\frac{\log (b)}{\log (a)}\right\rfloor =\left\lfloor \frac{\log \left(b^n+1\right)}{\log (a)}\right\rfloor$

Inspired by this question, I'd like to know how one would go about proving the below more general equation? $$n \in \mathbb{N},\;a \in \mathbb{N},\;b \in \mathbb{N}$$ $$b^n+1 \notin ...
5
votes
3answers
95 views

Solve $2^x+7=y^2$ for integer $(x,y)$

How many ordered solutions $(x,y)$ are there to the equation $2^x+7=y^2$ , where $x$ and $y$ are integers? I tried taking cases for $x$ and $y$ like they are even or odd but I couldn't solve ...