Question

For each positive integer $K$, let $S_k$ denotes the increasing arithmetic progression of positive integers whose first term is $1$ and whose common difference is $K$. For example $S_3$ is the sequence $1,4,7,10,.....$ For how many values of $K$, $S_k$ contains the term $2004$?

• A. $2$
• B. $8$
• C. $7$
• D. $1$

Answer 1

Answer: (A)

$2$

Series $S_k$ is $1,1+k,1+2k,...$ and so on
So $n_{th}$ term of series $S_k$ will be $1+(n-1)k$
for some n $n_{th}$ term would be 2004
So $1+(n-1)k=2004⟹(n-1)k=2003$
$n-1=\frac{2003}{k}$ should be an integer So k should be factor of 2003
So factors are $1,2003$ so there are only two factors for 2003
So two possible values for $K$ is $1,2003$
So answer is $2$