-1

enter image description here

I have a couple of issues with this question.

  1. If n has 1 digit then the difference is always 0, and 0 is not composite.
  2. If n includes 0, e.g. n=10, a permutation is 01. How do you interpret 01? As 1?

Need advice on proving the question.

Bill Dubuque
  • 272,048
AtKin
  • 628
  • 4
    I assume you mean "non-trivial permutation". Hint: look at divisibility $\pmod 9$. – lulu Jul 29 '22 at 16:59
  • @lulu: Technically, if you consider 0 to be a composite number, then it's true for the trivial permutation as well. – Dan Jul 29 '22 at 17:05
  • 1
    0 is not a composite number. So 1 digit integers don't satisfy this...ryt? – Nandeesh Bhatrai Jul 29 '22 at 17:08
  • 2
    @Dan I don't see this as a big issue here. Either restrict to non-trivial permutations, or consider $0$ composite (as it is certainly not prime). Doesn't change the core argument. – lulu Jul 29 '22 at 17:10
  • @lulu please see this https://byjus.com/question-answer/zero-is-neither-a-prime-nor-a-composite-number-why/ – Nandeesh Bhatrai Jul 29 '22 at 17:11
  • Just like in the linked reversal case, permuting the digits doesn't alter the digit sum, so they remain congruent mod 9, so 9 divides their difference, cf. this answer in the linked dupe. – Bill Dubuque Jul 29 '22 at 17:13
  • @NandeeshBhatrai As I said, I think this is a non-issue here. the case of a trivial permutation is clear and non controversial. To simplify the problem, I think it's best to restrict to non-trivial permutations, or just treat the trivial permutations as a special case. – lulu Jul 29 '22 at 17:14
  • @NandeeshBhatrai the q/a you linked has very poor English ("the fundamental THEORY of prime numbers") has major errors ("...states that any number can be written as the product of two PRIME numbers") and can't be trusted to reflect the general consensus of the mathematics community at large. There are certainly reasons to consider $0$ one or another or even reasons to consider $0$ its own special category. – JMoravitz Jul 29 '22 at 17:22
  • When talking about primes in a generic arbitrary sense, we usually refer to the abstract definition from algebra of rings and talk about prime ideals. The zero ideal does qualify as being an ideal... even qualifying it to be a prime ideal (if in the definition it wasn't specified that it have a nonzero element). In the end though, we know to be wary around zero and to be ready to qualify statements and phrase questions such that we handle those cases correctly. This question didn't. – JMoravitz Jul 29 '22 at 17:25
  • @JMoravitz the aim was just to tell whether 0 is composite or not. Kindly refer to this: https://brilliant.org/wiki/is-0-prime/ – Nandeesh Bhatrai Jul 30 '22 at 04:32
  • $0$ is not composite. Only integers greater than $1$ are considered to be "prime" or "composite". "Non-prime" would be the correct terminology here. But since permuting the digits usually means doing it in a way resulting in a different number (at least in everyday language) this issue is in fact a minor issue. – Peter Jul 30 '22 at 09:56

1 Answers1

2

It's well-known that $n\equiv S(n)\pmod{9}$ for all positive integers $n$, where $S(n)$ is the sum of the digits of $n$. Let $n'$ be the result when digits of $n$ are swapped. Note that $n-n'\equiv S(n)-S(n')\equiv 0\pmod{9}$ because $S(n)=S(n')$, thus $n-n'$ is composite.

TheBestMagician
  • 2,864
  • 2
    Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Jul 29 '22 at 17:18
  • @BillDubuque My bad, didn't see the dupe. – TheBestMagician Jul 29 '22 at 17:20
  • It's trivial to find such dupes, e.g. this search. What search did you try that failed? (we do expect answerers to search before answering, esp. for obvious exercises that are likely to have been asked before in the 12 years the site has existed). – Bill Dubuque Jul 29 '22 at 18:24
  • I didn't search for it, sorry. I'll keep that in mind in the future. – TheBestMagician Jul 29 '22 at 18:34
  • 2
    @BillDubuque A question: I've noticed that you (among others) are able to close a duplicate while citing multiple copies of it. That often comes up, but I don't see how to do it. I sometimes use the comments, but I'd prefer to do it your way. How is it done? – lulu Jul 30 '22 at 10:02
  • @lulu Iirc SE enables dupe list editing only for users who have a gold badge in one of the question's tags. – Bill Dubuque Jul 30 '22 at 13:51
  • @BillDubuque Yes, you are right. here is a reference. So at least I ought to be able to do it for probability. Thanks for the response. – lulu Jul 30 '22 at 13:56