Question

For each positive integer k, let Sk denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is k. For example, S3 is the sequence 1, 4, 7, 10...... Find the number of values of k for which Sk contain the term 361

Answer 1

Answer on Question #56759 – Math – Combinatorics | Number Theory

For each positive integer $k$, let $Sk$ denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is $k$. For example, S3 is the sequence 1, 4, 7, 10,...

Find the number of values of $k$ for which $Sk$ contains the term 361.

Solution

Since $S_k$ is increasing arithmetic sequence of integers whose first term is 1, then $a_n$ which belongs to $S_k$ can be represented as $a_n = 1 + kn$ (in this case $1 = a_0$), where $n$ is integer. Thus, there exists an integer $p$ such that $a_p = 1 + kp = 361$, hence $kp = 361 - 1 = 360$.

Let's find possible values of $k$. Since $k$ and $p$ positive integers (it is obvious that $p \neq 0$) and $360 = kp$, then $k$ and $p$ are divisors of 360, and $360 = 1 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5$.

Thus, $k = 2^i \cdot 3^j \cdot 5^t$, where $0 \leq i \leq 3$, $0 \leq j \leq 2$, $0 \leq t \leq 1$, $i, j, t$ are integer. Therefore, we obtain that there are $4 \cdot 3 \cdot 2 = 24$ different combinations of integers $i, j, t$. Thus, the number of values of $k$, for which $S_k$ contains the term 361, is 24.

Answer: 24.

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