# Tagged Questions

This tag is for questions about divisibility, that is, determining when one thing is a multiple of another thing.

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### Division problems

I came across these problems : 1) Find the lowest natural number $k$ that satisfies the condition : $7 \mid A$ , where $A = 194^{19} + 125^{14} + k$ 2) Find the different prime numbers ...
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### Is $a=\frac{1992!-1}{3449\times 8627}$ a prime number?

Is $a=\dfrac{1992!-1}{3449\times 8627}$ a prime number ? This is a natural follow-up to that recent MSE question We know that $a$ has $5702$ digits and no prime divisor $<10^6$.
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### Density of primes in intervals less than primorial numbers

Looking at the interval of the natural numbers $[1, p_{n}$#$]$; $\frac{1}{2}$ of the elements of this set will be even, and $\frac{1}{2}$ will be odd. $\frac{1}{3}$ of the elements of this set will ...
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### divisibility gcd

I was given this question below in class today but I'm unsure on how to do it and where to start. We learnt about this in class today but it was with numbers rather than letters so it has thrown me ...
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### Argue by contradiction : $n\in \mathbb N \to \; 4|(5^{n}-1)$ [closed]

I was working for various method to solve this :$n\in \mathbb N \to \; 4|(5^{n}-1)$ now I want to solve it only by "Argue by contradiction"
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### Show divisibility by 7

I was stuck at this question: Suppose $a^2+b^2=c^2$ for $a,b,c \in \mathbb Z$, and neither $a$ nor $b$ is a multiple of 7. Show that $a^2-b^2$ is a multiple of 7 I tried to write $b^2$ as ...
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### Probability of getting a five digit number divisible by 5 but with no two consecutive digits identical

A five digit number is written down at random. What is the probability of getting a number that is both divisible by 5 and doesn't have any 2 consecutive digits identical? I tried to analyse the ...
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### A question about the divisibility of certain polynomial sequences.

$2n+1=(n+1)^2-(n)^2$ . Therefore $(n+1)^2-n^2$ never divides $2$ for any integers.Can we make a similar statement for $(n+1)^x-n^x=a_n$ ... And if we can, can we combine polynomials to give us a ...
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### Form of Divisors of Proth numbers

Proth number is a number of the form : $z⋅2^k+1$ where z is an odd positive integer and k is a positive integer such that : $2^k>z$ Is there a form for divisors of Proth Numbers? (Like Mersenne ...
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### Divisibility question

Prove: (A) sum of two squares of two odd integers cannot be a perfect square (B) the product of four consecutive integers is $1$ less than a perfect square For (A) I let the two odd integers ...
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### Divisibility theory help

If $a$ is odd, show that $32 \mid (a^2 + 3)(a^2 + 7)$ Since $a$ is odd, I let $a = 2b + 1$ and did the expansion to get $16\mid [(b^2 + b +1)(b^2 + b + 2)]$, but I was unable to continue from ...
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### Probability number is divisible by half the square of a prime?

Let $p$ be a prime. What is the probability that a number of the form $\left \lceil \frac{p^2}{2} \right \rceil$ divides a random positive integer $n$. I have a solution that involves the Riemann-Zeta ...
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### Frazzle game question

In $7^{th}$ grade, in order to learn divisibility, memory, and focus, my math teacher had my pre-algebra class play a game called Frazzle. To play the game Frazzle, each person went around the room ...
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### Example of binary GCD for complex integers?

I know you can use bit shifting to speed up the GCD algorithm for a pair of integers. Is there a way to apply this idea to gaussian integers?
$a$ is an integer such that: $$a \mid \gcd(b_1,b_2,\ldots,b_z)$$ and $z$ can be very large. Does the GCD approach $a$ as $z$ grows? If yes, what is the relation between $z$ and $a$? Thanks...
### Can we always write $gcd(x,y)$ as $ax+by$ in UFD?
Let $R$ be a commutative ring with unity. Now assume that $R$ is Unique Factorization Domain, but not necessarily Principal Ideal Domain. Question: Let $x,y\in R$ be such that their GCD exists in ...