# divisibility

1. ### Divisibility

How many numbers are divisible by {1,2,..,n} inside interval [1,2n] ?
2. ### Find all a such that n^aâˆ’n is divisible by aâ‹…(aâˆ’1) for any integer n.

 $$a \cdot (a-1) \ \mid \ n^{a}-n \ \ \ \forall \ n \in \mathbb{Z}$$ Find all $a$. 
3. ### Divisibility proof

How to show that \; 11^{n} -6\; is divisible by 5 Without induction
4. ### Divisibility rule

If 3 divides n for n \in N Show that the sum of digits of n must be divisible by 3
5. ### Test divisibility polynomials

Hello everybody, I am currently revising some math out of pure interest. I found some interesting books, unfortunately not always with solutions to the problems. A problem I am stuck with is the following: "Proof that the polynomial n^3 + (n+1)^3 + (n+2)^2 is divisible by 9". I do not...
6. ### Divisibility and Remainders

Hello. There are two problems I'd like to have step-by-step explanations for so I can better understand how to break these problems down: 1) Prove that the number n^3 - n is divisible by 24 for any odd n. 2) Using GCD (a,b) = GCD (b, a-b), find the GCD of (2^100 - 1, 2^120 - 1).
7. ### Divisibility Rules?

https://brilliant.org/discussions/thread/divisibility-rules-4/ Prove that if x is divisible by 3, then 2^x - 1 is divisible by 7
8. ### Divisors of 154 forms a lattice, ordered by divisibility.

Hi, Hi have the following exercise about lattices. Having the lattice of integer divisors of 154, $L_{154}$, ordered by divisibility: i) Draw Hasse Diagram of $L_{154}$; ii) Get, if exists, complements of each element in $L_{154}$; iii) Establish if the lattice is distributive; iv) Establish...
9. ### USAMTS -- Year 28 (2016-2017) -- Round 3 -- problem 5/3/28

Here are the proofs I promised ... The problem at hand is problem "5/3/28" from round 3 http://usamts.org/Tests/Problems_28_3.pdf of this year's USAMTS online competition. Since the submission deadline for round 3 was yesterday (January 3, 2017), the problem is no longer "live", so...
10. ### Divisibility of a binomial coefficient

I want to proof, why \frac {\binom {2n}{n} - 2}{n} always returns an integer if n is a prime, but I don't know how.. You can write it like the following but I don't see a beginning for a proof...: \frac {\binom...
11. ### Divisibility problem

Determine all pairs $(a, b)$ of positive integers such that $ab^{2} + b + 7$ divides $a^{2}b + a + b$. How does one attack this?
12. ### divisibility proof question

please see image attached. for part (i), i used an induction. I wasn't bothered by this part. For part (ii), i managed to do the first part. The second part involves proving that S is divisible by different things depending on the even/odd status of n, which i was also fine with. The...
13. ### Divisibility by 7 Test

Greetings: I was somewhat fascinated by this chart by which divisibility by 7 can be tested. Fascinated probably because I could not figure out why it works. Perhaps some of you mathematically gifted persons can explain it to me: https://www.youtube.com/watch?v=vxHrfIom-YA
14. ### divisibility question

Kelby notices that if 2|36 and 9|36, then 18|36 since 18 = 2 * 9. He also notices that 4|36 and 6|36 but 24 does NOT divide 36. How could you help him understand why one case works and the other does not? My attempt... In the situation x|n and y|n if x and y have a common factor other than...
15. ### Divisibility

True/False? Explain... If a counting number is divisible by 3 and 11, it must be divisible by 33. My attempt: I will use a|b to mean "a divides b" a few examples first... 3|66 and 11|66 and 33|66.........all ok 3|99 and 11|99 and 33|99.........all ok It appears that the...
16. ### Divisibility

Write down your favorite three-digit number twice to form a six-digit number (e.g. 587,587). Is your six-digit number divisible by 7? How about 11? How about 13? Does this always work? Why? (Hint: Expanded form) My attempt: I examined a few of these numbers and discovered they're all...
17. ### Divisibility

a=2k+1 k>0 a could be prime or composite. It does not matter. I know which number divide (2^a)-1 without factorizing it I can find it quickly! Can you find it ? I give you one year to solve it.
18. ### Divisibility

Find all positive integers $a$ such that $5^{a}+1$ it's a divisor of $9^{a+1}+1$
19. ### Proof regarding divisibility of factorial by composite numbers.

If N is a composite number greater than 4, prove that: $(N - 1)! \equiv 0(mod N)$
20. ### Divisibility test for b+n in base b

I know that for b+1 you can use alternating digit sums to test divisibility in base b. However can I use alternating digit sums to test divisibility by any number that is of the form b+n where b is the base and n is what you add to base b?