Tagged Questions
1
vote
2answers
98 views
Are the propreties of arithmetic unproven?
For example, the property which says that $$a(b+c)=ab+ac$$ This is very clear for integers, but is it actually provable for all real numbers (and complex maybe). Or the commutative property which says ...
1
vote
1answer
46 views
Looking for name of theorem: “rational $\Leftrightarrow$ fractional part terminates or repeats”
I am looking for the name of the theorem that says that a number $x$ is rational if and only if its fractional part terminates or repeats (where "fractional part" refers to the representation of $x$ ...
5
votes
4answers
121 views
How to factor 5671?
The other day I wanted to factor 5671 in my head. (It turns out to be $53\cdot107$, but I did not know this at the time.) I quickly ruled out the easy divisors, 2, 3, 5, 7, 11, and 13. At this point ...
2
votes
2answers
50 views
$\frac{1}{ab}=\frac{s}{a}+\frac{r}{b} \overset{?}{\iff}\gcd(a,b)=1$
$$\frac{1}{ab}=\frac{s}{a}+\frac{r}{b} \overset{?}{\iff} \gcd(a,b)=1$$
This seems almost painfully obvious because it is just $ar+bs=1$ in another form. This second form is the definition of ...
1
vote
2answers
52 views
how to show that a function$f$ is contained in all natural numbers?
Let $f:\mathbb{N}\times\mathbb{N}\rightarrow \mathbb{R}$ be defined by $f(a, b) = \frac{(a+1)(a+2b)}{2}$.
Carefully show that the image of $f$ is contained in $\mathbb{N}$.
5
votes
2answers
53 views
How do I append an integer to the left of another integer?
For example:
. is my append operator
f(x,y) = |x| . |y|
f(1,45) = 145
f(233,10) = 23310
f(8,2) = 82
f(0,1) = 1
This is a trivially easy problem to ...
3
votes
4answers
128 views
Prove that $2222^{5555}+5555^{2222}=3333^{5555}+4444^{2222} \pmod 7$
I am utterly new to modular arithmetic and I am having trouble with this proof.
$$2222^{5555}+5555^{2222}=3333^{5555}+4444^{2222} \pmod 7$$
It's because $2+5=3+4=7$, but it's not so clear for me ...
1
vote
2answers
33 views
Constrictions on A.P with factorials.
There are five numbers $(a_1,a_2,a_3,a_4,a_5)$, such that they are in Arithmetic Progression.
Given that $a_1$ and $a_2$ are factorials, is there a possibility that either $a_4$ OR $a_5$ is a ...
2
votes
1answer
67 views
Cubes, squares and minimal sums
I have trouble solving the following task: i need to find positive integers a and b such that
1) $a \neq b$
2) $ \exists c \in \mathbb{N} : ~ a^2 + b^2 = c^3$
3) $\exists d \in \mathbb{N}: ~ a^3 + ...
2
votes
1answer
63 views
Maximal sum of positive numbers
I'll be grateful for any help with the foollowing question. I think the solution must be easy enough but i haven't figured it out yet.
Let a and b be positive integers such that
1) $\exists c \in ...
2
votes
1answer
108 views
Don't understand casting out nines
Let n be a positive integer. If the sum of the digits of n is divisible by 9, then n is divisible by 9.
I got upto here,
...
3
votes
2answers
78 views
Number base conversion
How can I convert a number from one base, $b_1 \neq 10$ to another base $b_2 \neq 10$ without going through base $10$ i.e. $b_1\rightarrow 10 \rightarrow b_2$?
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. ...
3
votes
2answers
106 views
selecting an arbitary digit from an integer
Let us say I have an integer of an arbitrary length such as:
$209484250490600018105614048117055336$
Is there an elegant function which allows me to select the $n$-th digit such that:
$f(1) = 6$
...
3
votes
4answers
157 views
Is the length of a segment between 0 and 1 exactly 1?
What is the length of the line segment between points A and B on a number line, where A = 0 and B = 1? Is it exactly 1? Perhaps I am thinking about it in an incorrect manner, but it seems to me that ...
11
votes
4answers
294 views
How many powers of 2 are easy to double? [duplicate]
Possible Duplicate:
Is 2048 the highest power of 2 with all even digits (base ten)?
Numbers written in base $10$ are easiest to double when their digits lie in the range $0, \ldots, 4$, so ...
10
votes
1answer
241 views
Given a set of digits, what is the biggest number we can make using exponentiation - numberphile noodle quiz
The question is motivated by a question on a can of number noodles.
Each item is a digit between $0$ and $9$. Clearly, if you form a string and consider it to represent a base $10$ integer, then ...
1
vote
1answer
99 views
Prove that given any rational number there exists another greater than or equal to it that differs by less than $\frac 1n$
I am currently attempting to prove a claim in Hardy's Course of Pure Mathematics and am currently stuck. I was hoping that someone would be able to provide some assistance on how to go about this.
...
1
vote
1answer
235 views
Multiplication Table with a frame and picture of equal sum
Is there an $n \times n$ multiplication table such that if you form a border of width $k$ ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")?
...
1
vote
1answer
188 views
How to solve multiplication alphametics?
I am referring to puzzles like these, where every letter represents a unique number (0-9):
...
1
vote
2answers
289 views
How to solve this alphametic (verbal arithmetic)?
I know I can get the answer for this puzzle but I'm struggling to see how to solve it.
Every letter represents a different number (0-9):
...
1
vote
1answer
121 views
Perfect squares
Wonder whether anybody here can provide me with a hint for this one.
Is $c=1$ the only case in which the expression
$(c^2+c-1)(c^2-3(c-1))$
returns a perfect square?
2
votes
2answers
143 views
remainder problem based on 5 and 7
When a number is divided by 5 than remainder is 2 and when the same number is divided by 7 remainder is 4. What will be remainder be when the same number is divided by 35?
What is the concept ...
1
vote
0answers
78 views
Does the shifting square root method work for non-integer bases?
Under "methods of computing square roots", Wikipedia states that the digit-by-digit calculation method, of which the shifting $n^{th}$ root algorithm is a generalization, works for all bases, but the ...
4
votes
2answers
131 views
Solving simple congruences by hand
When I am faced with a simple linear congruence such as
$$9x \equiv 7 \pmod{13}$$
and I am working without any calculating aid handy, I tend to do something like the following:
"Notice" that adding ...
17
votes
2answers
273 views
Multiplication tables with all entries distinct
Let positive integers $\alpha$ and $\beta$ be given. It is easy to
find sets $A$ and $B$ of positive integers such that:
$|A|=\alpha$ and $|B|=\beta$
The set $P = \{ab : a\in A, b\in B\}$ contains ...
0
votes
0answers
96 views
When can $a^m-b^m$ divide $a^m+b^m$. where $a$, $b$, $m$ are natural numbers, $a\gt b$
When can $(a^m-b^m)$ divide $(a^m+b^m)$, where $a$, $b$, $m$ are natural numbers, $a
\gt b$.
I approached this way:
Let $(a,b)=d$ and $\frac{a}{A}=\frac{b}{B}=d$, so $(A,B)=1$ and $A>B$.
...
0
votes
3answers
178 views
Which is the biggest integer that divides all integers that are the product of three consecutive odd numbers?
I read this problem from a high-school-math-problems-calendar, and I'm solving them in my spare time just for the fun of it (what in math is not about the fun? =) ), but this little one it's been hard ...
2
votes
1answer
70 views
Is there a pattern for reducing exponentiation to sigma sums?
The other day I was trying to find a method for cubing numbers similar to one I found for squaring numbers. I found that to find the square of a positive integer n, just sum up the first n odd ...
0
votes
1answer
412 views
HCF, LCM, and remainders
I think this is kind of a lame problem after seeing the quality of questions on this site, but I couldn't find anything related to my question
Now the basic questions is as follows,
Q. Find the ...
2
votes
2answers
200 views
Rewriting repeated integer division with multiplication
In many programming languages, such as C and C++, integer division of positive numbers is defined by simply ignoring the remainder. $5 / 2 == 2$.
In general, is it true of positive integers $a$, $b$, ...
5
votes
2answers
180 views
How can I write an algorithm to perform the following calculation exactly? (references accepted)
Given natural numbers $N, K, m, C$, with $3^{m/3}K>C$, I want to be able to write an algorithm to exactly compute the number
$$ \left\lceil \log_3 \left(\frac{N}{3^{m/3}K-C}\right) \right\rceil $$
...
-3
votes
3answers
430 views
Product of consecutive even numbers
Prove that the product of three consecutive even numbers is a multiple of 8?
show This into as much detail as possible!
3
votes
1answer
82 views
How fast is a low denominator encountered, when using only mediants?
This question is (remotely) related to How to find a "simple" fraction between two other fractions?, but is not answered in that older post.
Let $f_1=\frac{a}{b}$ and $f_2=\frac{c}{d}$ be ...
3
votes
1answer
87 views
Is there a direct proof of this inequality between quotients of integers?
Let $\frac{a}{b}$ and $\frac{c}{d}$ be two reduced fractions with
$bc-ad > 1$ (and hence $\frac{a}{b} \lt \frac{c}{d}$) and
$a,b,c,d$ positive. It is well known that there are integers $u,v$
...
3
votes
1answer
131 views
Finding any digit (base 10) of a binary number $2^n$
I was doing some Math / CS work, and noticed a pattern in the last few digits of $2^n$.
I was working in Python, in case anyone is wondering.
The last digit is always one of 2, 4, 8, 6; and has a ...
1
vote
1answer
109 views
A rule to determine the crossed out digit
Lets take any integer, $z=abc\cdots$, form the sum of its digits, $a+b+c+\cdots$, subtract this from $z$, cross out any one digit from the result, and denote the sum of the remaining digits by $w$. ...
3
votes
5answers
418 views
Trick to find multiples mentally
We all know how to recognize numbers that are multiple of $2, 3, 4, 5$ (and other). Some other divisors are a bit more difficult to spot. I am thinking about $7$.
A few months ago, I heard a simple ...
2
votes
4answers
76 views
How many new digits can appear in a multiplication?
When adding two positive integers, the result is sure to have at most the same number of digits as the largest of the two terms, plus one.
What about multiplication? Can many more digits can the ...
0
votes
2answers
158 views
From any given twelve integers, two have a difference that is a multiple of 5
How will you show that from a set of twelve given natural numbers (arbitrary) you can always find two such that their difference is divisible by $5$?
1
vote
3answers
329 views
The last digit of $n^5-n$
What will be the last digit of
$$n^5 - n \bmod 1000,$$
where $n$ is a natural number and $m \bmod 1000$ is the remainder when $m$ is divided by $1000$.
1
vote
0answers
211 views
How do I calculate cost savings and profits in this example?
How are profits and cost savings calculated in this table?
Is there enough given data to calculate these two?
Here is what I ...
9
votes
5answers
626 views
Prove that all even integers $n \neq 2^k$ are expressible as a sum of consecutive positive integers
How do I prove that any even integer $n \neq 2^k$ is expressible as a sum of positive consecutive integers (more than 2 positive consecutive integer)?
For example:
...
4
votes
5answers
236 views
Proving that $ 30 \mid ab(a^2+b^2)(a^2-b^2)$
How can I prove that $30 \mid ab(a^2+b^2)(a^2-b^2)$ without using $a,b$ congruent modulo $5$ and then
$a,b$ congruent modulo $6$ (for example) to show respectively that $5 \mid ab(a^2+b^2)(a^2-b^2)$ ...
3
votes
2answers
342 views
Base system and divisibility
I have seen the following one. Please give the proof of the observation.
We know that, The difference between the sum of the odd numbered digits (1st, 3rd, 5th...) and the sum of the even numbered ...
28
votes
7answers
899 views
Bad Fraction Reduction That Actually Works
$$\frac{16}{64}=\frac{1\rlap{/}6}{\rlap{/}64}=\frac{1}{4}$$
This is certainly not a correct technique for reducing fractions to lowest terms, but it happens to work in this case, and I believe there ...
