Tagged Questions

Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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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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Find two numbers whose sum is 20 and LCM is 24

With some guess work I found the answers to be 8 and 12. But is there any general formula for this? Note that the question is asked to my nephew who is at 4th grade.
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Existence of two primes satisfying the given conditions

I want to know whether the equation $x^a-x=y^b-y$ has a solution or not satisfying the conditions that $x$ and $y$ are distinct odd primes, $a$ and $b$ are integers both greater than $1$.
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If $p \equiv 1 \pmod{N!}$ prove there are no primitive roots $\pmod p$, $g$, that are less than $N$ [duplicate]

I've recently been looking at different questions and proofs in my book and one eludes me for the 2nd day in the row. Let $N \geq 4$. Show that if $p$ is a prime such that $p \equiv 1 \pmod{N!}$ ...
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Pick's Theorem for plane triangles in $\mathbb{R}^{3}$?

Pick's theorem asserts that, given a simple lattice polygon $p$ in $\mathbb{R}^{2}$, if $I$ is the number of lattices inside $p$ and $B$ is the number of lattices on the boundary of $p$, then the area ...
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Set Theory problem with unique numbers

Let $A_0$ be the set {$1, 2, 3, 4$}. Let $A_{i+1}$ be the set of all possible sums which can be obtained by adding two numbers in $A_i$ , where the two numbers do not have to be different. How many ...
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For what reltively prime integers $a$ and $b$ does the expression $2ab$ an even numeric palindrome? [closed]

I currently doing some research on numeric palindromes. And I am stack with the problem: For what reltively prime integers $a$ and $b$ does the expression $2ab$ an even numeric palindrome? Since ...
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Monkeys Dividing Pile of Bananas

Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey ...
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Proving that there are at least $n$ primes between $n$ and $n^2$ for $n \ge 6$

I was thinking about Paul Erdos's proof for Bertrand's Postulate and I wondered if the basic argument could be used to show that there are more than $n$ primes between $n$ and $n^2$. Is this approach ...
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Primitive roots for primes (Burton's text book)

On page 157 of Burton's elementary Number theory ,I am not sure if I understand the following reasoning:"Because 3 is a primitive root of 31 , any integer that is relatively prime to 31 is congruent ...
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Largest superprime number

Call a number n a superprime if every consecutive block of digits in n is also a prime. For example, the number 3727 contains the blocks 37 and 727, which are prime, but it also contains the blocks 72 ...
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1000 numbers on a blackboard

The numbers $1, 2, …,1000$ are written on a blackboard, in some order. Between every pair of consecutive terms, the absolute difference of the two terms is written between them, and then all the ...
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Deleting a Digit

Find the number of 5-digit positive integers n that have the following property: If we delete any digit in n, then we get a 4-digit number which is always divisible by 7. I did a little bit of modular ...
Integer solutions to $7^a + 1 = 2^b$
I'm looking for integer solutions to the equation $7^a + 1 = 2^b$. There are two obvious answers: $a=0, b=1$ $a=1, b=3$ I believe that those are the only solutions, but I am unable to prove it. I ...