Divisibility Definition and 170 Threads

I saw someone discussing divisibility rules in another thread and would thought I would make a note that the divisibility rule of 9 of summing the digits to see if you end up with 9 is really a trick of the counting base you are using (base 10). In general, this divisibility rule applies to...

Homework Statement if a and b are odd integer, then 8 l (a2-b2) Homework Equations n/a The Attempt at a Solution if a=b, clearly, 8 l (a2-b2) if not, now, I'm not sure how to continue should i varies b, and make a fixed, then varies a, and make b fixed, is that really the...

In a set of integers from 250 to 380, inclusive, how many are multiples of both 2 and 7? Please tell me if I'm correct: floor((380-250+1)/(7*2)) = 9 And in general is it always the floor of [cardinality]/[LCM]?

Homework Statement prove the following if a is an odd integer, then, 24 l a(a2-1) (i'm not familiar with modulo yet, i think it can help, but let don't use it yet ;P) Homework Equations n/a The Attempt at a Solution i stumbled when using 2n+1=a for all integer n, because i will only get...

Homework Statement proof the theorem if a l b and b l a then a=+-b Homework Equations The Attempt at a Solution there exist integer p,q such that ap=b and bq=a, then I've no idea how i can relate it to a=+-b.. clue please T_T

Homework Statement If 5 divides m^2 + n^2 + p^2 , prove that 5 divides wither m, or n, or p. Homework Equations m,n,p are all integers The Attempt at a Solution I am having some major problems with this chapter on modular arithmetic. any help is much appreciated! modular arithmetic is...

Homework Statement Prove that every palindromic integer N in base 10 with an even number of digits is divisible by 11. Then prove that every palindromic integer in base k with an even number of digits is divisible by k+1. Homework Equations palindromic means the number reads the...

Homework Statement Let M_n= 2^(n) - 1 be the n-th Mersenne number. a) Show that, if m|n, then M_m|M_n b) Show that, if m<n and m does not divide n, then GCD(M_n,M_m) = GCD(M_m,M_r) where r is the remainder of n upon division by m c) Let m,n be arbitrary natural numbers, and let d =...

Determine all possible value(s) of a 8-digit base 10 positive integer having the form ABCDEFGH, where each of the capital letters denotes a different digit from 1 to 9, that satisfy each of the following conditions: (I) AB is divisible by 2, and: (II) BC is divisible by 6, and: (III) CD...

Dear All, I have to test if a binary number is divisible to 2^n - 1 where n is even. Is there a test available for binary numbers like to test a divisibility by 3. Thanks in advance...

Homework Statement Prove that for any n \in Z+, the integer (n(n+1)(n+2) + 21) is divisible by 3 Homework Equations A previously proved lemma (see below) The Attempt at a Solution I sort of just need a nudge here. I have a previously proven lemma which states: If d|a and d|b...

Homework Statement Let a,b be integers where a doesn't =0. Prove that if a divides b, and b divides a, then a=b or a=-bThe Attempt at a Solution I started out with b=aj and a=bk, where j,k are integers. Don't quite know how to proceed

Here is an interesting problem that I've been thinking about for a while: Let p be a prime s.t. p = 4m+1 for some integer m. Show that p divides n^2 + 1, where n = (2m)! It comes from a section on principal ideal domains and unique factorization domains. It is well-known that p is the...

Well i found this sentence: "If the number of even bits minus the number of odd bits is a multiple of 3 (e.g. -3,0,3,6, etc) then the number is divisible by three." Can anyone tell me the proof of this? Thanks

Hello, If we are given that b3|a2, how do we show that b|a? I started off looking at prime factorizations, but I could use a push in a more substantial direction.

Hi all, so I was looking at Legendre symbols, and I saw that \left(\frac{2}{p}\right)=(-1)^{\frac{p^2-1}{8}}. How does one show that \frac{p^2-1}{8} is always an integer? That is, how can we show that 8 | p^2-1? Can a similar method be applied to show that 24 | p^3-p? Thanks :-)...

Homework Statement Let's take a prime number p not equal to 5. Now let's take three integers a,b,c. Prove that if p | (a + b + c) \wedge p | (a^5 + b^5 + c^5), then p | (a^2 + b^2 + c^2) \vee p | (a^3 + b^3 + c^3) Homework Equations I think: (a + b + c)^2 = a^2 + b^2 + c^2 +...

if p is any prime other than 2 or 5, prove that p divides infinitely many of the integers 9, 99, 999, 9999, ... If p is any prime other than 2 or 5, prove that p divides infinitely many of the integers 1, 11, 111, 1111, ... Is there a way to do this problem using modular arithmetic? Thanks

Homework Statement A polynomial p(x) leaves the rest 3 when divided by (x+2) and the rest 8 when divided by (x-6). What's the rest r(x) when p(x) is divided by (x+2)(x-6)? Homework Equations The Attempt at a Solution I wrote the three equations: p(x)=q1(x+2) + 3 p(x)=q2(x-6) +...

Can you help me with this problem: if a^2 divides b^2, show that a divides b. This was a homework question that I had a while ago and it was solved by using the fundamental theorem of arithmetic. I instead tried to solve it with proof by contradiction: a^2 divides b^2 implies a divides...

How can I show that (a^n -b^n) doesn't divide (a^n + b ^n) for all integers a,b, and n? I have that if (a^n -b^n) did divide (a^n + b ^n), then (a^n +b^n) = q (a^n -b^n) which implies b^n = -q*b^n (mod a^n). Then 1 = -q (mod a^n), meaning gcd(b, a^n) = 1. I am unsure of what more I can...

First of all, I hope this problem is supposed to be here - I'm Swedish and in Sweden "calculus" & "precalculus" are rather odd terms. Anyway..Homework Statement Prove that n3 - n is divisible by 6 if n is a natural number, and divisible by 24 if n is an odd natural number.Homework Equations The...

Homework Statement a, b, P, and any other numbers introduced are members of the integer set. If P is known to be a prime number, and ab can be divided by P, then prove that either a or b can be divided by P. Homework Equations All properties of real numbers. Need not be explicitly...

Homework Statement A is a square matrix of size n, B is of size m, C is an m*n(typo,should be n*m) matrix and n>m ,Rank(C)=m. if AC=CB, prove characteristic polynomial of B divides that of A. Homework Equations nothingThe Attempt at a Solution I think I need to prove any eigenvalue of B is an...

P is a 8-digit base ten positive integer having the form ABCDEFGH that uses each of the nonzero digits from 1 to 8 exactly once, and satisfies all of these conditions: (i) AB is divisible by 8. (ii) BC is divisible by 7. (iii) CD is divisible by 6. (iv) DE is divisible by 5. (v) EF is...

\binom{n+1}{k+1}=\binom{n}{k}+\binom{n}{k+1} I'm not sure how to prove this. However...does this work: If p is a positive prime number and 0<k<p, prove p divides \binom{p}{k} Can't I just say that if that binomial is prime, this means that it is only divisible by p and 1 (since...

Can someone please tell me how to go about answering a question like this? I've been racking my brain for a long time and still don't have a clue...I guess because my background in algebra/number theory really isn't that strong. "What is the greatest integer that divides p^4 - 1 for every...

Homework Statement let a belong to N and x,r belong to Z use the definition of divisibility along with the axioms of Integers to prove that IF 5|a and 15|(2ax+r) then 5|r Homework Equations How do I continue the proof?? The Attempt at a Solution So I have: let a belong to N and...

which elements of Z/60 are invertible?? what are their inverses? do we have any quick way to do this kind of question??

Homework Statement Suppose that n^2 is odd and that 3 does not divide n^2. Show that 12|(n^2-1) Homework Equations none The Attempt at a Solution Well I know that since n^2 is odd, n^2-1 is even. I'm not sure what the next step would be.

Homework Statement Prove that any integer n >= 2 such that n divides (n-1)! + 1 is prime. Homework Equations The Attempt at a Solution I'm having trouble getting started, I have no idea how to approach this, can someone give a hint on where to begin maybe because I'm just not...

For example, is 5i "divisible" by 5? Or does divisibility only apply to integers? On that note, is 5pi divisible by 5? Is 5/6 not divisible by 5? Thanks in advance! =)

I was trying to work out whether or not 2n+3 divides (2n+1)! for positive integers n. After trying a few cases I think it does not work but I don't know how a proof for this would work. I tried induction but it got really messy. I also tried rephrasing it, such as putting it into modular...

given two integers A and B that are very big is there any 'fast' algorithm to calculate the remainder of the division \frac{A}{B} or in other similar words to say if B divides or not A thanks.

1) We know that if \theta is trisectible (with straightedge and compass), then \theta/3 is constructible. But is it also true that if \theta/3 is constructible, then \theta is trisectible (with straightedge and compass)? If so, then I can say that since 15o is constructible, we have that...

Homework Statement Prove that n^3 - n is divisible by 6, when n is a nonnegative integer. The Attempt at a Solution Mathematical induction: It works for n=0 It works for n=1 (Extra step, just in case) Check if it works for the (k+1)th step. For it to work, it must be expressible...

Homework Statement what is the test to to see if a number is divisible by 11 and prove it. The Attempt at a Solution If the alternating sum of a numbers digits is divisble by 11 then so is the number. I don't know how to prove it tho.

[SOLVED] larson 3.3.19b Homework Statement What is the largest number N for which you can say that n^5-5n^3+4n is divisible by N for every positive integer N. EDIT: change the last N to n Homework Equations The Attempt at a Solution I have just been plugging in things for n and seeing what...

Homework Statement Prove that if n^2+m and n^2-m are perfect squares, then m is divisible by 24.Homework Equations The Attempt at a Solution I found all of the squares mod 24. They are:{0,1,4,9,12,16}. We want to show that if we take anyone of these as n^2, then n^2+m and n^2-m cannot be in...

[SOLVED] induction proof Homework Statement Given a set of n+1 integers between 1 and 2n (inclusive), show that at least one member of the set must divide another member of the set. Use induction. Homework Equations The Attempt at a Solution When n=1, this is obvious. Assume the result is...

Homework Statement Let n be a natural number. If the number formed by the last three digits of n is divisible by 8, then n is divisible by 8. Homework Equations Natural numbers are the set of {1,2,3,4,5,6,...} The Attempt at a Solution I believe we should use a direct proof to...

Someone please help me with this qiestion: Prove that for all integers a, b, and c, if a divides b but not c then a does not divide b + c, but the converse is false. Thanks.

I need some help proving this statement. Prove that a positive integer is divisible by 5 if and only if it's last digit is either 0 or 5. Thanks

So I'm taking an introductory number theory course as an undergraduate, and this particular "genre" of questions really just has me stumped. Pick a prime p such that p is odd. Now, take various sums up of natural numbers from 1 to p, and show that the results are divisible by p. For...

Find all prime numbers (p,q,r), that numbers pq+pr+rq and p^3+q^3+r^3-2pqr are divided by p+q+r

We have recurrent sequence of integer number a_{1},a_{2},... a1=1, a2=2 a_{n}=3a_{n-1}+5a_{n-2} for n=3,4,5,... Is integer number k>=2, that (a_{k+1}*a_{k+2}) mod a_{k} = 0 ? Please for quick help :)

Show that for any n>1 we can construct a positive integer consisting of only 1's and 0's such that this integer is a multiple of n.

Homework Statement Use Mathematical Induction to prove that 12^n + 2(5^{n-1}) is divisible by 7 for all n \in Z^+ Homework Equations The Attempt at a Solution First, show that it works for n = 1: 12^1 + 2 \cdot 5^0 = 14 , 14/7 = 2 Next assume: 12^k + 2(5^{k-1}) = 7A Then, prove for...

Is 0 an odd or even number? The reason why I ask is this: I need to write cosh(x) as the sum of an even and odd function. I could only come up with cosh(x) = cosh(x) + 0, where cosh(x) would be the even and 0odd. However, this doesn't make any sense since 0 is exactly divisible by 2 with no...

If 3p^2 = q^2 and p and q are integers, how do I prove that 3 is a common divisor for p and q? My attempt: q^2 is divisible by 3, so q is divisible by 3. I can't prove that p is divisible by 3.