# Tagged Questions

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### The largest integer less than $n$ is $n-1$

Let $n$ be a positive integer. Prove that the largest integer which is less than $n$ is $n-1$. Attempt of a solution: Since $n$ is a positive integer $n>0$. I think I have to use the well-ordering ...
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### How to proof this: (m is odd ∧ n is odd)⇒ m + n is even

I don't quite understand why I can not proof the following: Assume that n,m ∈ N. Show: (m is odd ∧ n is odd)⇒ m + n is even. With this: Say n, m are odd. Then the remains of (m + n) / 2 is equal to ...
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### My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
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### Induction question help.

Let $x$ and $y$ belong to a commutative ring $R$ with prime characteristic $p$. Show that, for all positive integers $n$ $$(( x + y )^p)^n = (x^p)^n + (y^p)^n$$ I hope you can can understand ...
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### Divisibility proof by induction.

$169$ | $3^{3n+3}-26n-27$ ? Fulfilled for $n=0$. Induction to $n+1$: An integer $x$ exists so that $169x= 3^{3n+6}-26n-27-26$ $169x= 27*3^{3n+3}-26n-27-26$ $169x= 26*3^{3n+3}+3^{3n+3}-26n-27-26$ ...
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### My first proof employing strong induction / complete induction (very simple number theory). Please mark/grade.

What do you think about my first proof employing strong induction? What mark/grade would you give me? Theorem Every natural number greater than 1 is a product of one or more primes. Proof First, ...
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What do you think about my first induction proof? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof First, introducing a predicate ...
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### Prove that all elements in the set T is divisible by 3 using structural induction.

Let T be the set defined recursively as follows: (1) (0,3) $\in$ T (2) If (x, y) $\in$ T, then (x + 2, y - 1) $\in$ T and (x - 3, y) $\in$ T. (3) Every element of T can be obtained from (1) by ...
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### When is there an $m$ that divides $u^{an+b}+v^{cn+d}$ for all $n$

This is a generalization of Prove by induction? which asks how to prove that $73$ divides $8^{n+2}+9^{2n+1}$for all $n$. Here is my generalization: Find conditions on positive integers $u, a, v, c$ ...
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### Is this divisibility proof by induction correct/sufficient?

To show: 13 | $4^{2n+1}+3^{n+2}$ I used induction beginning successfully with n=0 (or n=1), then making the step to n+1: An x exists so that $13x = 4^{2n+3}+3^{n+3}$ $13x = 16*4^{2n+1}+3*3^{n+2}$ ...
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### Proving divisibility by using induction: $133 \mid (11^{n+2} + 12^{2n+1})$ [duplicate]

If $n > 0$, then prove the following by using induction: $$133|(11^{n+2} + 12^{2n+1}).$$
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### Prove by induction that for all $n$, $8$ is a factor of $7^{2n+1} +1$

I want to prove by induction that for all $n$, 8 is a factor of $$7^{2n+1}+1$$ I have proved it true for the base case and assumed it true for $n=k$, but when I cannot figure out when to go towards ...
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### I'm not sure if my subscripts are lining up correctly in this elementary number theoretic induction proof

First, the motivation for the below lemma is to use in a proof that every number has a unique representation in a base. My question is that when using the inductive hypothesis, I'm not sure if my ...
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### Proving $r!$ divides the product of r succesive positive integers

I have to prove the following theorem: Prove that the product of $r$ consecutive positive integers in divisible by $r!$ I am having a hard time getting a generalization down for the full set of ...
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### Proving that $fib(n) < (5/3)^n$ for $n \ge 1$ by induction

I know this has been shown before here but no post really answered my question. I had this problem given to me as an induction practice problem and I couldn't solve it without help. When I got the ...
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### Proving the base case for a problem in elementary number theory

I have a question about how to prove statements such as the following, using induction: If $p \mid a_1a_2 \cdots a_k$, then $p \mid a_i$ for some $i$, $i = 1, 2, \ldots, k$, where $p$ is prime. ...
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### Prove by induction that $2^{2n} – 1$ is divisible by $3$ whenever n is a positive integer.

I am confused as to how to solve this question. For the Base case $n=1$, $(2^{2(1)} - 1)\,/\, 3 = 1$, base case holds My induction hypothesis is: Assume $2^{2k} -1$ is divisible by $3$ when $k$ is a ...
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### I've got a small problem with induction

Let me take a quick example: We want to prove by induction that $3^n-1$ is a multiple of 2, where n is a positive integer. So we start with our "base case" and show that $3^1-1$ is indeed a multiple ...
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### How Can I find the summation of divisors of $n^p$.

For Example $n=8$ and $p=2$. So $n^p=64$. And the summation of divisors is $1+2+4+8+16+32+64=127$. But the problem arises when $n=10^6$ and $p=10^6$. Remember u can modulus the result by $100$.
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### Stuck while trying to prove $2k^3 \geq (k + 1)^3$…

how can I prove the following: $2k^3 \geq (k + 1)^3$ This is the final part of the elaborate proof for $2^n > n^3$ give $n \geq 10$ I have used induction and end up with: $2^{K+1} > 2k^3$ ...
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### Divisibility Of Positve Integers [closed]

Suppose a,b and c are three positive integers which satisfy the condition that ($a$2+$b$2+$c$2) is divisible by $(a+b+c)$. Prove that there exists infinitely many positive integers $n$ for which ...
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### Frobenius coin problem, 5 and 9

I am hoping to get some help with two problems related to Forbenius coin problems. A) A fictional government has decided to issue currency in only 5 and 9 value denominations. Prove that there is a ...
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### Prove by induction that $3\mid (n^3 - n)$

I'm having an argument with my professor whether my exam was right or not. Before I sign a formal complain to get a review on my exam, I'd like to be sure it's correct. My answer: Proof by induction: ...
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### Demonstration that $\forall\; n>3,\;\;n^2<n!$

How do I prove by mathematical induction that$$\forall\; n>3,\;\;n^2<n!$$ I tried, $n=4$ then $4^2<4!$ what is true, because $16<24$.Hypotesis: $n^2<n!$ Thesis: ...
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### How can I show that $4^{2n}-1$ is divisible by $15$ for all $n$ greater or equal to $1$

Ok so this is a question from a book that has no included solution and I think I'm on the right way but I just need a little help. The question is: Show, for all $n \ge 1$ such that $4^{2n} - 1$ ...
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### fill-in-the-blank induction proof

I'm stuck at homework task I'm working on. I would really appreciate some help. Here is the task: Let $f$ be a function on binary numbers defined recursively as follows. $f(0) = 1$ and ...
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### How can I prove by induction that $9^k - 5^k$ is divisible by 4?

Recently had this on a discrete math test, which sadly I think I failed. But the question asked: Prove that $9^k - 5^k$ is divisible by $4$. Using the only approach I learned in the class, I ...
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### Induction and Countable Set

Ok well everytime ive seen induction being used, its been on the naturals for a statement we wish to prove. My question is would any countable set also work? Hence, doing induction on the rationals as ...
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### Prove some divisibility results by induction

Please hint me, I have two questions: Prove by induction that: 1) $${13}^n+7^n+19^n=39k,\,\, n\in\mathbb O$$ in which $\mathbb O$ is the set of odd natural numbers. 2)  ...
### Given $n \in \mathbb{N}$ prove that a polynomial result gives a natural number.
A friend asked me this question: Prove that for every $n\in \Bbb N$ the next equation result: $\dfrac{n^3}{6}+\dfrac{n^2} {2}+\dfrac{n}{3}$ would be a natural number. My instincts were that i need ...
Let $a,n\in\mathbb{N}$, show that there exists $m\in\mathbb{N}$, such that $(a+1)^n=ma+1$ I tried to do by induction on $n$, but found it a bit strange the demonstration. ...