Difference between revisions of "2010 AMC 12A Problems/Problem 20"
(Created page with '== Problem 20 == Arithmetic sequences <math>\left(a_n\right)</math> and <math>\left(b_n\right)</math> have integer terms with <math>a_1=b_1=1<a_2 \le b_2</math> and <math>a_n b_n…') |
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| − | Hence, we have to find the largest <math>n</math> such that <math>\frac{a_n-1}{n-1}</math> and <math>\frac{ | + | Hence, we have to find the largest <math>n</math> such that <math>\frac{a_n-1}{n-1}</math> and <math>\frac{b_n-1}{n-1}</math> are both integers. |
Revision as of 21:46, 11 February 2010
Problem 20
Arithmetic sequences 
and 
have integer terms with 
and 
for some 
. What is the largest possible value of ?
Solution
Since and have integer terms with 
, we can write the terms of each sequence as
where 
and 
are the common differences of each, respectively.
Since
it is easy to see that
.
Hence, we have to find the largest such that 
and 
are both integers.
The prime factorization of 
is 
. We list out all the possible pairs that have a product of
and soon find that the largest 
value is 
for the pair 
, and so the largest value is 
.
