Divisibility Definition and 166 Threads

Homework Statement Let n be any odd integer. Prove that 1 is the only "common" divisor of the integers n and n+2. The Attempt at a Solution I don't think I understand the question. The few notes I have state d| (n+2 )- n This resembles n+2 ##\equiv## n mod d , but I don't see the...

So I think I've just proven a preposition, where ##0## is divisible by every integer. I prove it from the accepted result that ##a \cdot 0 = 0## for every ##a \in \mathbb{Z}##. From then, we can just multiply the result by the inverse of ##a##, to show that the statement holds for ##0##. That is...

Given unequal integers $a, b, c$ prove that $(a-b)^5+(b-c)^5+(c-a)^5$ is divisible by $5(a-b)(b-c)(c-a)$.

Hey, I've been stuck on these questions for awhile. They're bonus/ extra practice questions and I have a midterm coming up and I'm not quite comfortable with the process. If anyone can help me that'd be great! Prove the following theorem: for all integers a, b and c, if a does not divide b - c...

Show that for every n∈N, 34n+2 +1 is divisible by 10 Prove by Induction. Attempt) Base Case: n = 1, 3(4(1)+2) + 1 = 730 So the base case holds true. Assume that the inequality holds for n = k 34k+2 +1 is divisible by 10 Show true for n = k+1 34(k+1)+2 + 1 34k+4+2 + 1 34 * 34k+2 + 1 81 *...

Hi all, I am trying to figure out if there is a pre-existing theorem and proof of whether or not each of the binomial coefficients in a binomial expansion of (a +b)^n are divisible by n, particularly in the case where n is a prime number. Has this already been asked and answered somewhere in...

Find all 10-digit numbers ##a_1a_2a_3a_4a_5a_6a_7a_8a_9a_{10}## (for example, if ##a_1= 0##, ##a_2## = 1, ##a_3 = 2## and so on, we get the number 0123456789), such that all the following hold: - The numbers ##a_1,~a_2,~a_3,~a_4,~a_5,~a_6,~a_7,~a_8,~a_9,~a_{10}## are all distinct - 1 divides...

Homework Statement Prove that if gcd(a, 133) = 1, then 133 divides (a^18 - 1). The Attempt at a Solution This is an old homework question as I'm going over the homeworks to review for the test, but can't seem to get this right. Which is annoying because I remember I did it fine back in the...

If a divides b, and a divides b+c then a divides 3c. How do I go forward with this? This is what I've done so far: suppose a|b and a|b+c then b = an for some integer n and c = am for some integer m ∴ b+c = an+am = a(n+m) = ak for some integer k but I...

Homework Statement Show that in any set of 172 integers there must be a pair whose difference is divisible by 171. Is the result true if the word difference is replaced by sum. I think it should say distinct integers The Attempt at a Solution I think I should start by partitioning...

I think my answer is correct, i just need some peer review. Homework Statement Let k, m, n ∈ Z+ where k and m are relatively prime. Prove that if k|mn then k|n The Attempt at a Solution This question seems trivial. We know the property that if x|y then x|yz for all integers z. Therefore...

Homework Statement Prove that for any n ∈ Z, n(n² − 1)(n + 2) is divisible by 12 . The Attempt at a Solution We first assume n = k for some value k. Next we assume k(k² − 1)(k + 2) = 12m for some value m. I don't know where to go from here. I don't think this is supposed to be an induction...

is this true or false: If a|b and a|c, then one (or both) of b|c or c|b holds. if I want to disprove this, can I: let a = 5, x = 2 and y = 3. b=ax c=ay then c=bz and c = bg doesn't hold.

Let a, b and c be positive integers such that a^(b+c) = b^c x c Prove that b is a divisor of c, and that c is of the form d^b for some positive integer d. I'm not sure how to solve this question at all, I need some help.

Hello, The following problem appears in my number theory text: The answer: I have tried to trace the reasoning in reverse. I understand how we get to the finish (by showing that the number is divisible by all of the relatively prime factors of n, but I don't understand how we...

Can anyone help me confirm if I have solved this correctly? Many thanks. Homework Statement Q. 9^n-5^n is divisible by 4, for n\in\mathbb{N}_0 The Attempt at a Solution Step 1: For n=1... 9^1-5^1=4, which can be divided by 4. Therefore, n=1 is true... Step 2: For n=k...

I posted this in the number theory forum to no success... so I figured maybe the homework help people would have some input Let x,y,z be integers with no common divisor satisfying a specific condition, which boils down to 5|(x+y-z) and 2*5^{4}k=(x+y)(z-y)(z-x)((x+y)^2+(z-y)^2+(z-x)^2) or...

the problem of infinite divisibility and how QE sheds some light. QE = quantum entanglement A quanta of energy is considered the smallest possible energy "unit" in the universe. note to readers: the below is hypothetical and could have errors ...of various kinds i have asked this to be moved...

1.To prove - For any natural number n, the number N is not divisible by 3 2. N = n2+1 3. Dividing naturals into three classes according to remainder outcomes during division by 3 ie. 0,1,2 ; for any whole number k ---> 3k, 3k+1, 3k+2 And then substitute the values respectively to...

Chinese remainder theorem help Homework Statement Solve in Z^{2}:6x-5y=1 Conclude the solution to the system: X≡2(mod5) , X≡1(mod6) The Attempt at a Solution 1- solved the equation and found one unique solution which was S={(1,1)} Given: X≡2(mod5) , X≡1(mod6) X≡2(mod5) means X=5t+2...

Homework Statement Show that if p is an odd prime of the form 4k + 3 and a is a positive integer such that 1 < a < p - 1, then p does not divide a^2 + 1 Homework Equations If a divides b, then there exists an integer c such that ac = b. The Attempt at a Solution We have to do this proof by...

The Order Divisibility Property states that if an = 1 (mod p), then the order ep(a) of a (mod p) divides n. How can I go about proving this? Additionally, if a is relatively prime to p, when does the congruence am = an (mod p) hold? Is there a proof for this as well? Thanks!

Figured it out nvm

Homework Statement Prove: If a|b and b|c then a|c. Assume a, b and c are integers. Homework Equations none The Attempt at a Solution If a divides b then that means that there is a real integer "r" that is ra=b . and since we assume b divides c then c=bs. After...

Modulus problem[solved] I would like to know how to prove that the number n^3 + 2n is divisible by 3 for all integers n. I know that I am supposed to examine the possible remainders after division with 3. The possibilities are: n= 0,1,2 (mod 3) Could somebody please help me? How can the...

Homework Statement I'm trying to figure out how many successive 1's are necessary for a number composed solely of 1's to be divisible by another number x. For example, how many 1's are necessary for 1...1 to be divisible by 7? Simply performing the calculation shows that the first such...

Homework Statement Prove by induction that no matter how one chooses a set of n+1 positive integers from the first 2n positive integers, one integer in the set divides another integer in the set. 2. The attempt at a solution Tried direct induction. Base case easy to prove. P(n+1) is with n+2...

Homework Statement Q. Prove by induction that... (please see attachment). Homework Equations The Attempt at a Solution The end result should be divisible by 6, but hasn't worked out for me. Can someone help me spot where I've gone wrong? Thank you.

Homework Statement If two integers a and b are relatively prime and if each divides an integer n, prove that their product ab divides nHomework Equations 1=sa+tb for some integers s and t (thm 1.35) gcd(a,b)=1 n=aa'=bb' The Attempt at a Solution I have tried many many different ways to...

Find the largest natural number m such that n^3 -n is divisible by m for n element N. Prove your assertion. How exactly do you begin this im thinking the largest m could possible be is n^3 -n, but I am not sure. To be divisible we know that mk=n^3-n for some k

Homework Statement Prove the following: If 5 divides a^2 + b^2 + c^2 then 5 divides a and 5 divides b and 5 divides c. Homework Equations 5 \mid a \implies a=5k , k \in Z The Attempt at a Solution My idea is to assume 5 divides a^2 + b^2 +c^2 also assume that "5 does not divide a"...

Does anyone know the answer to this problem: if you have a set of consecutive prime numbers (2,3,5,7...), what is the greatest amount of consecutive integers that are divisible by at least one of the prime numbers? For 2,3,and 5, I know it is 5 (2 through 6, 24 through 28, 32 through 36...), but...

Hi, we know that for all interger m, n_1!\cdots n_k! divides m! where n_1+\cdots+n_k=m. Now I want to show that m! (n!)^m divides (mn)!. We see that (n!)^m divides (mn)! since \overbrace{n+\cdots+n}^{m-terms}=mn Also m!(nm-m)! divides (mn)! similarly. But how could I show my required...

Homework Statement determine which are true and false. 4|0 and 0|0 Homework Equations a is divisble by b provided there is an integer c such that bc=a The Attempt at a Solution on the first one 4|0 4x=0 , 0 would work for this. and 0x=0 any integer would work for this but I...

1. For any integer n, prove that 3 divides n^3 -n The Attempt at a Solution I'm stuck. I understand that means that n^3 -n mod 3 =0. or I can n^3 -n can be expressed as 3x. But I don't know how to prove it. Where do i go from here. Thanks

Homework Statement Question: If n|a^2 and n|b^2 prove or disprove that n|ab. I think this is true, but am having trouble proving it. Homework Equations a^2=nm, b^2=nk a^2-b^2=n(m-k) a^2+b^2=n(m+k) The Attempt at a Solution Essentially I've tried all sorts of algebraic...

CONJECTURE: Subtract the Absolute Values of the Stirling Triangle (of the first kind) from those of the Eulerian Triangle. When row number is equal to one less than a prime number, then all entries in that row are divisible by that prime number. Take for instance, row 6 (see below). The...

Homework Statement Given that n is a positive integer, prove by induction that (a^n-b^n) is divisible by (a-b) Homework Equations n = k n = k+1 The Attempt at a Solution a^(k+1) - b(k+1) = (a-b)A, where A is a positive integer. I am lost from here or not sure if this is even the...

Hi everybody! I have this problem: Either P = (X+2)m+(X+3)n and Q = x2+5x+7; Determine m, n such that Q | P;( m, n = ? (Q divide P)); May you help me please? Thank You!

Homework Statement Prove that if N=abc+1, then (N,a)=(N,b)=(N,c)=1. Homework Equations The Attempt at a Solution Assume N=abc+1. We must prove (N,a)=(N,b)=(N,c)=1. Proceeding by contradiction, suppose (N,a)=(N,b)=(N,c)=d such that d\not=1 . Then we know, d | N and d | abc. Thus...

Homework Statement I have to prove or disprove the following: Part a) If p is prime and p | (a2 + b2) and p | (c2 + d2), then p | (a2 - c2) Part b) f p is prime and p | (a2 + b2) and p | (c2 + d2), then p | (a2 + c2) Homework Equations The Attempt at a Solution Part a)...

Let f(x), g(x) be in F[x]. Show that if g(x)|f(x) and f(x)|g(x), then f(x)=kg(x) for some k in F. Since g(x)|f(x), then f(x)=g(x)r(x) for some r(x) in F[x]. Similarily, since f(x)|g(x), then g(x)=f(x)s(x) So f(x)=f(x)s(x)r(x)

Question: If gcd(a,42)=1, show that a^6 - 1 is divisible by 168. Answer: So I know that if 42 were prime, than the Little Fermat Thm says that a^p-1 is congruent to 1 mod p. But I have no idea where to start if p is not prime. Help please.

Homework Statement let a and b be relatively prime positive integers. if c is a positive integer and a | bc then prove that a | cThe Attempt at a Solution I started by trying to prove the contra-positive. If a is not divisible by bc then a is not divisible by c. it follows that a mod bc > 0...

Can someone help me with the following proof. prove that if n is a positive integer then n^7 - n is divisible by 7. This should be done by breaking it down into 7 cases.

Homework Statement i want to show if n = pq, 1 < p < n, then p l (n-1)! Homework Equations n/a The Attempt at a Solution i can see its true, because p < n, p l p, then p l (n-1)!. and this prove very ambiguous for me 2 question. 1.help me, i think there must be easier way to prove the...

{SOLVED}Number theory/ divisibility Show that m^2 is divisible by 3 if and only if m is divisible by 3. MY attempt: I assumed that 3k=m for some integers k and m. squared both sides and now get. 3n=m where n=3*(3k^2). Thus 3|m^2 Now the problem is when i assume: 3k=m^2 and need...

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]?