Difference between revisions of "2021 Fall AMC 10B Problems/Problem 22"
Arcticturn (talk | contribs) (→Solution 1) |
Arcticturn (talk | contribs) (→Problem) |
||
Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
For each integer <math> n\geq 2 </math>, let <math> S_n </math> be the sum of all products <math> jk </math>, where <math> j </math> and <math> k </math> are integers and <math> 1\leq j<k\leq n </math>. What is the sum of the 10 least values of <math> n </math> such that <math> S_n </math> is divisible by <math> 3 </math>? | For each integer <math> n\geq 2 </math>, let <math> S_n </math> be the sum of all products <math> jk </math>, where <math> j </math> and <math> k </math> are integers and <math> 1\leq j<k\leq n </math>. What is the sum of the 10 least values of <math> n </math> such that <math> S_n </math> is divisible by <math> 3 </math>? | ||
+ | |||
<math>\textbf{(A)}\ 196\qquad\textbf{(B)}\ 197\qquad\textbf{(C)}\ 198\qquad\textbf{(D)}\ 199\qquad\textbf{(E)}\ 200</math> | <math>\textbf{(A)}\ 196\qquad\textbf{(B)}\ 197\qquad\textbf{(C)}\ 198\qquad\textbf{(D)}\ 199\qquad\textbf{(E)}\ 200</math> | ||
Revision as of 11:22, 23 November 2021
Problem
For each integer , let
be the sum of all products
, where
and
are integers and
. What is the sum of the 10 least values of
such that
is divisible by
?
Solution 1
To get from to
, we add
.
Now, we can look at the different values of mod
. For
and
, then we have
. However, for
, we have
Clearly, Using the above result, we have
, and
,
, and
are all divisible by
. After
, we have
,
, and
all divisible by
, as well as
, and
. Thus, our answer is
. -BorealBear
Solution 2 (bash)
See Also
2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.