Question

What is the least positive integer n such that n^2 - n is divisible by some but not all integer values of k when 1=< k =< n?

Answer 1

1

Answer 2

k=1, and n=1

Answer 3

\[1=<(k=1)=<(n=1)\] and if n is squared, \[n^2=1^2=1\]

Answer 4

@TheJoker but 1^2-1 = 0

Answer 5

where did the subtraction come from?

Answer 6

"What is the least positive integer n such that n^2 - n "

@TheJoker

Answer 7

oops... didn't see the -n :P

Answer 8

then n can equal 2

Answer 9

and so can k if you want

Answer 10

@TheJoker it said the answer was 5:

Note that $n^2-n=n(n-1)$ is divisible by $1$, $n-1$, and $n$. Since we want $n^2-n$ to be divisible by some but not all integer values of $k$ when $1\le k\le n$, we must have $n-1>2$ so $n>3$. If $n=4$, $n$ is divisible by 2, so $n^2-n$ is divisible by all integer values of $k$ when $1\le k\le n$. Therefore, the least $n$ is $n=\boxed{5}$.

Answer 11

then why did you ask here then? lol

Answer 12

@TheJoker
No its Alcumus on AoPs... its if u get the answer wrong then itll give the solution once ur wrong... it lowered my score lol

Answer 13

ok