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For each positive integer K, let S_{k} denotes the increasing

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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 $2004$ ?

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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

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Text solutionVerified

Series

S_{k}

1, 1 + k, 1 + 2 k, ...

and so on
So

n_{t h}

term of series

S_{k}

will be

1 + (n - 1) k

for some n

n_{t h}

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

1, 2003

So answer is

2

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Question Text	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 $2004$ ?
Answer Type	Text solution:1
Upvotes	150

For each positive integer K, let Sk​ denotes the increasing arithmetic progression of positive integers whose first term is 1 and whose common difference is K. For example S3​ is the sequence 1,4,7,10,..... For how many values of K, Sk​ contains the term 2004?

Text solutionVerified

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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 $2004$ ?