# Tagged Questions

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### Let $x = 2441921$. Factor $x$ into a product of primes.

Let $x = 2441921$. Factor $x$ into a product of primes. I found that: $1519^2 −x=−134560= −2^5 ·5 · 29^2$ and $1541^2 −x=−67240= −2^3 · 5 · 41^2$. I am trying to figure out what to do from here. ...
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### Topic = Numbers ,{Simple But difficult for me :) } [on hold]

Question= There are how many different "a" natural number ?
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### The sum of two irrational square roots

This is very similar to this question, but I was wondering if there was a simpler proof. In particular, a proof that would prove that $\sqrt{x}+\sqrt{y}$ is an irrational number if both $\sqrt{x}$ ...
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### How does this method work? [closed]

Let $n=16$ for an example: step 1: get set of prims from $1$ to $\sqrt{2n}: \{2, 3, 5\}$, step 2: get set of $n \mod 2, n \mod 3, n \mod 5: \{0, 1, 1\}$, setp 3: from $0$ to $n-3$, ...
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### Need help finding a number x so that $\phi > 9x/10$?

I need help finding a number $x$ so that $φ(x) > 9x/10$? ($φ$ being Euler’s phi function.) I also need to find a number $x$ so that $φ(x) < x/3$?
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### Suppose that $m \ge 0$ show that $49 \mid 5\cdot3^{4m + 2} + 53\cdot2^{5m}$

I've re-written the equation in a few different ways hoping for a few different approaches: $$49y = 5 \cdot 3^{4m + 2} + 53 \cdot 2^{5m}$$ I think the first equation has more potential, since it ...
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### $7^n-1$ is divisible by $6$ for all natural number $n$ [closed]

How to prove $7^n-1$ is divisible by $6$ for all natural number $n$. Thanks for your help.
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### Is 2(2k-1) is a perfect square for positive integer k?

For positive integer $k$, let $M = 2(2k-1)$, which of the following must be true? (a) $M$ is not a perfect square for any $k$. (b) There are infinitely many $k$ such that $M$ is a perfect square. ...
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### Find the natural numbers $n$ in which $n^2$ divides $584$? [duplicate]

I'm trying to find the natural numbers $n$ in which $n^2$ divides $584$ ? i tried all the ways i know but i get stuck.
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### Given perimeter of triangle and one side, find other two sides

In triangle ABC, all three sides have integer lengths. If AB = 21, the perimeter is 54, and the area is a positive integer, what are the lengths of BC and AC? I tried using Heron's Formula, but I ...
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### Sum of Digits Question

If A is the sum of the digits of $5^{10000}$, B is the sum of the digits of A, and C is the sum of the digits of B, what is C? I know it has something to do with mod 9, but I'm not sure how do use it ...
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### Let $a$ and $b$ be coprime positive integers. Prove that, for any integer $n$, there exist integers $s$ and $t$ such that $sa + tb = n$

I always sort of took this fact for (well..) fact. Can someone help me with the proof? Does this question have something to do with modulus? Since $a$ and $b$ are coprime ($gcd$ = 1), multiplying ...
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### Number theory problem exercise? [closed]

Find all natural numbers $N$ so that $\varphi(N)=24$ where $\varphi$ is Euler's function.
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### Help with $\sum_{d\mid n}τ(d)^2=\sum_{d \mid n}τ(d)^3$

I am doing some exercises on number theory on multiplicative number theoretic functions and I have some problems with the multiplication on sums like the sum $\sum_{d\mid n}(τ(d))^2$ where $d$ is a ...
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### Application of Dirichlet Theorem in AP to elementary number theory problems.

I have learnt this theorem in my class, however, "elementary" examples are very limited. (focusing more on analytic machinery) Are there any interesting applications to elementary number theory that ...
The two forms are: $\ 3x^2 + (6y-3)x - y\$ $\ 3x^2 + (6y-3)x + y - 1, \ \ x,y \in \mathbb{Z}^{+}$ For example: $\ \ \ 5 = \ 3*1^2 + (6*1-3)*1 - 1\$ ,when $\ x = y = 1\$,of the two forms \$\ ...