Divisibility Definition and 166 Threads

I need to prove that if n is a natural number and n^2 is divisible by 6, then n is divisible by 6. I know that I knew how to do this at one point fairly recently, if you could refresh my memory I would greatly appreciate it.

Hi, I am going through the book on number theory by ivan niven. well its tough book though, and i am stuck with problems in the first topic divisbility.Hope some help. 1. prove that a|bc if and only if a/(b,c)|c where (b,c) is the lcm of b and c. 2. Prove that there are no positive...

HEllo everyone. I'm trying to find a counter example that will prove this false. But it may be true but I'm hoping it isn't :) For all integers a and b, if a|10b then a|10 or a|b. I said false, a = 3, b = 5. 3 is not divisible by 50. 3 is also not divisble by 10 nor 3 divisible by 5...

I'm having trouble with this question, I need to prove or disprove this statement: If c divides (a+b), but c does not divide a, then c does not divide b. what i have so far is ck = (a+b) where k is some integer. Next I have a=ck-b and b=ck-a. I tried doing things like a = ck-(ck-a) but...

Hello, I'm struggling with the question on induction. I was wondering if you could help me? Prove that n(n^2 +5) is divisible by 6 for n belonging to Z^+ P_1 is (1(1^2 + 5))/6=1 hence P_1 is true If P_k is true then (k(K^2 +5))/6=r and if and only if (k(k^2 +5))=6r then P_(k+1) is...

The question is: Prove by mathematical Induction that f(n) \equiv 2^{6n}+3^{2n-2} is divisible by 5. This is what I did: Suppose that the given statement is true for n=k Since the f(k) is divisible by 5, f(k)=5A (where A are is a constant.) Also, from the given statement...

Two questions here. I know the definitions, but cannot formulate a through proof. 1.a and b are positive integers. If a^3 | (is divisible by) b^2, then a | (is divisible by) b. Now, by definition, I know that a^3*k=b^2, for some k. Also, I know that a * j = b for some j. But where do...

Where \mathbb{Z}^{+} represents the set of all positive integers, How do I prove that \begin{gathered} \forall \left\{ {a_0 ,a_1 ,a_2 , \ldots ,a_n } \right\} \subset \mathbb{Z}^ + \;{\text{where}}\;\max \left\{ {a_0 ,a_1 ,a_2 , \ldots ,a_n } \right\} \leqslant 9, \hfill \\ \left(...

I have 2 questions. 1)what is the remainder with 100! is divided by 103? explain your answer 2)a = 238000 = 2^4 x 5^3 x 7 x 17 and b=299880 = 2^3 x 3^2 x 5 7^2 17. Is there an integer so that a divides b^n? if so what is the smallest possibility for n? the first one i have no...

Hello I would like to know how to determine whether a number is divisible by 11 or not... say 209. I know the divisibility rules from 2 to 10... but how about 11?

I am reading Naoki Sato's notes on Number Theory: http://donut.math.toronto.edu/~naoki/nt.pdf I am on page 2, and doing Example 1.1... "Let x and y be integers. Prove that 2x+3y is divisible by 17 iff 9x+5y is divisible by 17." There's ALREADY A SOLUTION on the book, but I do not...

How would I go about proving that 8^n - 3^n (n >= 1) is divisible by 5 using mathematical induction? I tried this but I do not think it is right: First, prove that 8^1 - 3^1 is divisible by 5. 8^1 - 3^1 = 5, which is divisible by 5. Second, prove that 8^(k+1) - 3^(k+1) is divisible by 5 if...

We've seen divisibility testing for different numbers...for 2,3,4,5,7,11... But now I want to know is there any way to find divisbility with any prime number? Suppose, how can I say 12783461236 is divisible with 97 or not?

Divisibility of "c" by "a "and "b" but not "ab". Hello, I am having trouble with this question: i) Give an example of three positive integers a,b,c such that a|c and b|c but ab does NOT divide c. ii) In the situation of part (i), is there a condition that guarentees that if a|c and...

Hello, I am supposed to prove or disprove this statement: Let m,d,n,a be non-zero integers. If m = dn, and if m|an, then d|a. I had a proof but I made an error. Stay tuned for my revised proof. Ok! Here is my corrected proof: By definition: An integer "a", is a...

The thread “Sorites Heap Paradox” brought the following thought to mind. Quantum mechanics may not answer the infinitely divisible question in the way that one would expect. Normally we think of discrete units here, but consider this: I read that at the University at Boulder [I think] an...