Tagged Questions
5
votes
3answers
110 views
Prove that $(a+1)(a+2)…(a+b)$ is divisible by $b!$ [duplicate]
The problem is following, prove that:
$$(a+1)(a+2)...(a+b)\text{ is divisible by } b!\text{ for every positive integer a,b}$$
I've tried solving this problem using mathematical induction, but I ...
1
vote
2answers
149 views
Understanding mathematical induction for divisibility
I'm on my quest to understand mathematical induction proofs (beginners). First, thanks to How to use mathematical induction with inequalities? I kinda understood better the procedure, and practiced it ...
18
votes
5answers
1k views
Why $a^n - b^n$ is divisible by $a-b$?
I did some mathematical induction problems on divisibility
$9^n$ $-$ $2^n$ is divisible by 7.
$4^n$ $-$ $1$ is divisible by 3.
$9^n$ $-$ $4^n$ is divisible by 5.
Can these be generalized as
$a^n$ ...
2
votes
3answers
95 views
Proving that an expression divides a number
How do you prove that
$$n(n+1)(n+2)$$
is divisible by 6 by using the method of mathematical induction?
According to my book
$$\begin{aligned}
(n+1)(n+2)(n+3) &= n(n+1)(n+2)+3(n+1)(n+2)\\
&= ...
6
votes
6answers
1k views
Show that $11^{n+1}+12^{2n-1}$ is divisible by $133$.
Problem taken from a paper on mathematical induction by Gerardo Con Diaz. Although it doesn't look like anything special, I have spent a considerable amount of time trying to crack this, with no luck. ...
-1
votes
3answers
464 views
Induction principle for verifying divisibility
I am stuck on one question: Show that $8^n-3^n$ is divisible by $5$.
Thank you
1
vote
2answers
130 views
Prove that $2^n | P(2n, n)$
I am attempting to use Induction to prove this, but I am not sure if it is the right method to take. Here is what I have tried:
Induction Hypothesis:
Assume $P(k)$ is true for some fixed $ k \geq 1$ ...
2
votes
3answers
357 views
Prove by induction: $2^n + 3^n -5^n$ is divisible by $3$
Let $P(n) =
2^n + 3^n - 5^n
$.
I want to prove that $P(n)$ is true for all integers $n\geq 1$.
The basis step for this proof is easy enough: $P(1)$ is divisible by $3$.
For the inductive step, I ...
3
votes
1answer
127 views
Good resource of maths problems with solutions
I'm searching for a good book or web page that has a good amount of problems and their solutions, at undergraduate level, of divisibility, inequalities, induction, etc.
Thanks in advance
