4. Determine the common terms of the arithmetic progressions:

$$
1,5,9,13,17,21, \ldots \quad 4,15,26,37, \ldots
$$
Solution. Let $a_{n}$ and $b_{m}$ be the general terms of the arithmetic progressions; then we have:

$$
a_{n}=1+4(n-1), \quad b_{m}=4+11(m-1)
$$

We need to solve the equation $a_{n}=b_{m}$ in the set of natural numbers. This equation reduces to solving the equation

$$
11 m=4(n+1)
$$

from which we get $n+1=11 k, m=4 k$. Therefore, we have $a_{11 k-1}=b_{4 k}$.