If we have $n$ consecutive integers, then one of these integers is divisible by $n$. Prove the above statement.
Prove that one of $n$ consecutive integers must be divisible by $n$
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1@MartinSleziak silly me! – 2012-04-01
5 Answers
HINTS: Let the integers be $m,m+1,\dots,m+n-1$. If $m$ is a multiple of $n$, you’re done. If not, write $m=qn+r$, where $q$ and $r$ are integers, and $0
Is $m+c$ one of the $n$ consecutive integers? In other words, is it true that $0\le c
? Is $m+c$ divisible by $n$? Why?
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0@EthanAlvaree: Thanks. Yes, that second $m$ was a typo. (0
is right, though.) – 2015-04-16
There are $n$ residue classes mod $n$. If you have $n$ consecutive integers, they will fill up all $n$ residue classes. One of them must be in the residue class of things divisible by $n$; i.e. things $=0\pmod{n}$.
Proof by induction.
For all $i \in \mathbb{N}_{\ge 1},$ let $P(i)$ be the proposition: $ \exists q \in \mathbb{N}_{\ge 1} \text{ such that } qn \in \{i, i+1, \ldots, i+n-1\}. $
Base case: $P(1)$ is trivial since $n \in \{1, 2, \ldots, n\}.$
Induction hypothesis: Assume $P(i)$ holds for some $i > 1.$ In other words, there exists some $q$ such that $qn \in \{i, i+1, \ldots, i+n-1 \} \tag{1}$
Induction step: Consider the set $\{i+1, i+2, \ldots, i+n \}.$ The induction hypothesis in equation $(1)$ implies two cases:
Case I: $i \neq qn,$ i.e. $qn \in \{i+1, \ldots, i+n-1\}.$ Hence $qn \in \{i+1, i+2, \ldots, i+n \}.$ So $P(i+1)$ holds.
Case II: $i = qn,$ and hence $i+n = n(q+1).$ Since $i+n \in \{i+1, i+2, \ldots, i+n \},$ we have that $P(i+1)$ holds.
We have just shown that $P(i) \implies P(i+1).$
QED.
Hint $\ $ Let $\rm\ a\ $ be the largest in the sequence. $\rm\: mod\ n\!:\ a\equiv r\,\in\, [\:\!0,\:\!n)\:$ so $\rm\:n\ |\ a\!-\!r\,\in\, [\!\:a,\:\!a\!-\!n)$
This has widespread applications, e.g. this proof that $\rm\:n\:|\:(n\!-\!1)!\:$ for composite $\rm\:n.$
Note $\ $ This is just a shifted equivalent of the Division Algorithm, i.e. the following equivalence is true in any euclidean domain $\rm Z\:$ (domain with an algorithm for division with 'smaller' remainder)
$\begin{array}{rcrlrl} &\rm \exists\!\!\!&\rm q,r \in Z, &\rm\ a\: =\: q\:n + r, \!\!\!\!&\rm |r| \!\!\!&\rm <\ |n| \\ \displaystyle\rm{b\: =\: a\!-\!r\ \atop{\huge\iff\atop\phantom{M}}} &\rm \exists\!\!\! &\rm q,b\in Z, &\rm\ b\: =\: q\:n, \!\!\!\!&\rm |a\!-\!b| \!\!\!&\rm <\ |n| \end{array}\qquad$
\ let \ k,k-1,k-2,....,(k-(n-1)) \ are \ 'n' \ consecutive \ integers.
$\begin{equation} \begin{aligned} k(k-1)(k-2)(k-3).....(k-(n-1))=& \frac{k(k-1)(k-2)(k-3).....(k-(n-1))(k-n)....1}{(k-n).....1}\\ =&{\frac{k!}{(k-n)!n!}.n!}\\ \end{aligned} \end{equation}$ $\implies n!|k(k-1)(k-2)....(k-(n-1))$ $\implies n|k(k-1)(k-2)....(k-(n-1))$
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0This is incomplete/wrong. For instance $4$ divides $6 \times 2$, but does not divide either $6$ or $2$. – 2012-04-06