Questions on basic arithmetic involving numerical quantities only. Questions involving variable values (other than the result of the operation) should be placed under the (algebra-precalculus) tag.

learn more… | top users | synonyms (1)

11
votes
6answers
231 views

Simplifying nested square roots ($\sqrt{6-4\sqrt{2}} + \sqrt{2}$)

I guess I learned it many years ago at school, but I must have forgotten it. From a geometry puzzle I got to the solution $\sqrt{6-4\sqrt{2}} + \sqrt{2}$ My calculator tells me that (within its ...
2
votes
2answers
35 views

Greatest Common Divisor of a+b,a-b

Prove that $gcd(a + b, a − b) ≥ gcd(a, b)$ Let $d=gcd(a+b,a-b)$ So $d=m(a+b)+n(a-b) = a(m+n)+b(m-n)$ Which implies $d|a$ and $d|b$ Therefore, $d|gcd(a,b)$ $gcd(a,b)=dx ≥ d= gcd(a + b, a − b)$ Why am ...
0
votes
0answers
98 views

Divide and Conquer division algorithm explained

I am trying to understand the divide and conquer algorithm that is used in the GMP bignum arithmetic library. The code is very optimised and that makes it somewhat hard to understand. the doc does ...
3
votes
1answer
60 views

Fibonacci sequence digits

We define the Fibonacci sequence by $F_{n+2}=F_{n+1}+F_{n}$, with $F_0=0$ and $F_1=1$. How to compute the last $30$ digits of $F_{2^{200}}$ for instance? can we use Python?how?
-2
votes
1answer
21 views

Result by dividing both side of equation

Let $C(2^n) = 2C(2^n-1) + 2^n$. According to a book, when dividing both sides by $2^n$ it gives: $\dfrac {C(2^n)} {2^n} = \dfrac {C(2^{n-1})} {2^{n-1}} + 1$. The only thing which I don't understand ...
1
vote
1answer
51 views

Greatest Common Divisor of natural numbers

Suppose two natural numbers a, b satisfy ab = n for some fixed integer n. What is the maximum possible value of gcd(a, b)? Let $d= gcd (a,b)$ So, $d= xa + yb$ I don't know how to proceed with this. ...
0
votes
3answers
25 views

How is countably infinite addition defined

In the axiom of additivity of probability theory, the concept of a countably infinite sum, i. e. the sum of countably infinitely many real numbers, is used. Could someone please tell me how that kind ...
1
vote
1answer
24 views

What does “proportional” mean?

I used to think of $\propto$ as indicating the one quantity is proportional to the other, with possibly an additive constant involved, i.e. $f(x) \propto g(x)$ if $f(x) = ag(x) + b$. Is that ...
4
votes
1answer
54 views

Write formulas in specific languages of group.

So, for each of the following groups write a formula in the language of group theory, which holds in given group, but doesn't hold in others two. $(i)$ The integers with addition \ I think it's ...
6
votes
4answers
176 views

Why is $\sqrt {12} = 2 \sqrt 3$?

Why $\sqrt {12} = 2 \sqrt 3$? It is obvious? If we considered the function $f(s) = s^2 $ it is injective on positive numbers so we obtain the conclusion. But in the same time it is an equality ...
1
vote
1answer
42 views

Determine what when multiplied with $180$ gives a perfect cube

Recently, at a math competition, I was given the following question: Determine the smallest number that gives a perfect cube when multiplied by $180$ . I had thirty seconds to solve this question and ...
-2
votes
1answer
100 views

Kindergarden help arithmetic problem [closed]

What is 20 + 25? I can't seem to count with my fingers... I only have 10 :( Mummy and daddy won't help me:(
0
votes
2answers
32 views

Why not a simple division in this case?

n represents total number of persons. max represents the maximum number of persons by department. To know the accepted number ...
1
vote
2answers
30 views

Inequality problem

$5\geq X \geq 2$ $0.8 \geq Y\geq 0.5$ Find range of the values of A) $\sqrt{XY}$ B) $(\frac{X}{Y})^2$
0
votes
4answers
88 views

Fastest way to perform arithematic calculations

This is one of the questions asked for Junior Trader position at prop. trading firm. Perform the following operation. 4.3 * 0.58 + 2.0E-5 - 0.9 How can one ...
3
votes
1answer
45 views

Partitioning the natural numbers to finite number of arithmetic progressions

Some already asked on this site whether the natural numbers can be partitioned to finite number of arithmetic progresions with distinct differences, with the condition that the intercection between ...
2
votes
1answer
52 views

What is the area of the square given the following circumstances?

The perimeter of square HJKL is 2 times the perimeter of square WXYZ The SO if perimeter of HJKL is 2 times the perimeter the WXYZ than $$ \text{Perimeter of } HJKL = 2\cdot2(l+w)$$ So the L and ...
4
votes
3answers
57 views

How to solve the following arithmetic radical problem?

$$ 2(4\sqrt{7} + 1 + 3\sqrt{7} + 2) $$ I distribute first right? $$ 8\sqrt{14} + 2 + 6\sqrt{14} + 4$$ $$ 14\sqrt{14} + 6$$ BUT IT LOOKS LIKE ITS SUPPOSED TO BE $$14\sqrt{7} + 6$$ I also have a ...
0
votes
0answers
19 views

Calculating inflation using CPI having m-o-m and y-o-y data

inflation Hello guys, I am having trouble calculating inflation from CPI - I would like to know what was the CPI in the yellow box... so far I have not been able to solve this... I would be extremely ...
1
vote
1answer
85 views

Prove that $(n!)!$ divisible by $(n!)^{(n-1)!}$ [duplicate]

I was trying to think of a situation and use combinatorics to solve the problem. Any other arithmetic solution is also appreciated.
2
votes
3answers
78 views

How to derive that for every real $y > 0$, for every positive real $z \neq 1$, there is a $x \in \mathbb{R}$ such that $y=z^x$.

I am not sure on how to derive the following statement concerning the reals (that I think should be true). For every real $y > 0$, for every positive real $z \neq 1$, there is a $x \in ...
-10
votes
3answers
136 views

$-25+55+(85+65)= 180$ or $120$ ?And Why? [closed]

I mean this question can have both the anwers but which one is correct and why? $-25+205=180$ And $-30+150=120$ But which answer is correct ?
0
votes
1answer
26 views

Find original cost based on fractional purchase

How would I go about finding the original cost of bitcoin knowing that $20 purchased .0531401 of bitcoin? I would like to know what the cost of 1 bitcoin was at the time of purchase? ...
9
votes
9answers
26k views

What's the formula for the 365 day penny challenge? [duplicate]

You might have seen the viral posts about "save a penny a day for a year and make $667.95!" The mathematicians here already get the concept while some others may be going, "what"? Of course, what the ...
-4
votes
3answers
137 views

BODMAS riddle - which is correct? [closed]

I fully understand the order of BODMAS - It's the order of operations in maths equations. So lately this little puzzle/riddle has been going around on social media - ...
1
vote
2answers
46 views

Struggling with a task with brackets

The task is to change brackets (operations, numbers and their positions should be remained the same) to make $F$ equal to $850$. $F = (1 + 2) \times (3 + 4) \times (5 + 6) \times (7 + 8) \times (9 ...
0
votes
2answers
37 views

How to solve a word problem when given width and height of the following?

The width of a room is 4 feet shorter than its length, and its height is 3 feet less than its length. The area of four walls is larger than the sum of the areas of the floor and ceiling by 134 square ...
2
votes
1answer
70 views

Follow-on to exponents in Birthday Paradox

Follow on to discussion on Birthday Paradox I read the Scientific American article again Please explain why they want to find the probability of NOT matching birthdays, and then subtracting that ...
2
votes
2answers
337 views

Not understanding division in Birthday Paradox

I am reading Scientific American's explanation for birthday paradox here I understand everything in the article up to Every one of the 253 combinations has the same odds, 99.726027 percent, of ...
1
vote
1answer
60 views

Prove that two arithmetic grammars generate the same language

In my university's automata theory book it is claimed that the following two arithmetic grammars generate the same arithmetic language but no proof is shown. $G_1=(\{E\}, \{a,+,*,(,)\}, \{E\to ...
3
votes
2answers
91 views

Show $\frac{(p^d - 1)(p^{d-1} - 1)}{(p-1)(p^2 - 1)} \equiv 1 \pmod{p}$.

Let $p$ be prime and $d \ge 2$. I want to show that $$ \frac{(p^d - 1)(p^{d-1} - 1)}{(p-1)(p^2 - 1)} \equiv 1 \pmod{p}. $$ I have a proof, but I think it is complicated, and the statement appears in ...
0
votes
1answer
47 views

Binary subtraction with borrowing vs. 2's complement

Consider the following two binary numbers which are using sign-magnitude representation. Therefore, both are positive numbers. ...
3
votes
3answers
93 views

What is value of $a+b+c+d+e$?

What is value of $a+b+c+d+e$? If given : $$abcde=45$$ And $a,b, c, d, e$ all are distinct integer. My attempt : I calculated, $45 = 3^2 \times 5$. Can you explain, how do I find the distinct ...
2
votes
6answers
103 views

Prove that $5^n + 2\cdot3^{n-1} + 1$ is multiple of $8$

Prove that $5^n + 2\cdot3^{n-1}+ 1$ is multiple of $8$. I've tried using induction (it isn't): For $n=1$: $$5^1 + 2\cdot3^{n-1} + 1 = 8$$ If it is true for $n$, then $n+1$? \begin{align} 5^{n+1} ...
37
votes
4answers
2k views

Property of 111,111

Whilst playing on my calculator, I noticed the following pattern. $1^2-0^2=1$ $6^2-5^2=11$ ${20}^2-{17}^2=111$ ${56}^2-{45}^2=1{,}111$ ${156}^2-{115}^2=11{,}111$ To me, this is where it gets ...
-3
votes
2answers
105 views

Can $60+60\times0+1$ be both $1$ and $61$ [closed]

The expression under debate is $60+60\times0+1$. I'm in a debate on Facebook and people that are saying $61$ and that the people that are saying $1$ are and vice versa but I'm saying they are both ...
-2
votes
1answer
33 views

Numbers $65x1y$ multiples of 12 [closed]

Find all the five digit numbers in the form $65x1y$ multiples of $12$
2
votes
1answer
33 views

How could I simplify this algebra expresion?

I had been solving an equation with complex numbers: $z = \frac{x - iy}{x + iy}$ I solved it up to the point where I get: $z = \frac{x^{2} - y^{2}}{x^{2} + y^{2}}$. But I have no idea how to simplify ...
1
vote
2answers
58 views

Ratios with A.P.

Two A.P.’s have the same number of terms. The ratio of the last term of the first progression to first term of the second progression is equal to the ratio of the last term of the second progression ...
2
votes
1answer
56 views

How do I calculate $1.496\,\text{E}11$? [closed]

Sorry for that noobie question but how do I calculate this type of number $1.496\,\text{E}11$?
5
votes
2answers
56 views

Arithmetic: Prove that is multiple of 30

Prove that $n^{19}-n^7$ is multiple of $30$ I've seen $6$ can divide it because $$n^{19}-n^7=n^7(n^{12}-1) = n^7(n^6+1)(n^6-1)=n^4(n^6+1)(n^3-1)n^3(n^3+1)$$ And there are three consecutive ...
2
votes
1answer
50 views

How to find $a^b$, where $a$ and $b$have more than $10$ digits?

Consider any two numbers $a$ and $b$ of more than 10 digits, how to find $a^b$ (without the aid of computing devices). Is there any shortcut method to do it. other than binomial series. How do I solve ...
6
votes
1answer
91 views

Do there exist integers $a,b,c$ such that $a^5+b^5+c^5=2016abc$ and $a+b+c=5776$?

This question should be solvable without a calculator - I tried playing around with odd/even properties, but didn't get very far. I also tried looking at the average of $a,b,c$ (about $1900$), but ...
2
votes
1answer
45 views

How do you calculate the exponent of an exponent

How do you calculate the exponent of an exponent? In what order do you calculate the exponents? For example, to calculate ${2^3}^4$ Is it $({2^3})^4 = 8^4$ or $2^{3^4} = 2^{81}$ ADDED: Say ...
0
votes
1answer
30 views

If $\gcd(a_i, a_j)=1$ for $i\ne j$ then $\gcd(a_2 … a_n, …, a_1 … a_{n-1})=1$?

I try to prove this statement for $i,j \in \{1,...,n\}$. I notice that for each term for instance $a_1$ divides $(n-1)$ terms and it is the same for each $a_k$ but I think it is not enough to ...
0
votes
1answer
32 views

How to invert matrix in finite field

I want to invert matrix $A$ in the finite field $\mathbb{F} = \mathbb{F}_2[x]/p(x)\mathbb{F}_2$ with $p(x)=x^8+x^4+x^3+x+1$. This finite field is used by the encryption scheme AES. $A = ...
0
votes
1answer
58 views

Least number principle

I'm really sorry if this question has been asked before, I looked but couldn't find anything. The least number principle states that every non-empty set of positive integers contains a least element. ...
0
votes
2answers
102 views

Find the number to replace the question mark in between two pairs of numbers

Here is the problem: I have to find the number to replace the question mark. I know there is a series or a pattern to find it any hint will be very helpful.
0
votes
1answer
30 views

Division in finite fields

Let's take $GF(2^3)$ as and the irreducible polynomial $p(x) = x^3+x+1$ as an example. This is the multiplication table of the finite field I can easily do some multiplication such as ...
0
votes
1answer
106 views

Proof of Euclid's Lemma in N that does not use GCD

I am looking for a proof of Euclid's Lemma, i.e if a prime number divides a product of two numbers then it must at least divide one of them. I am coding this proof in Coq, and i'm doing it over ...