20. Find $p$, if each of the numbers $p, p+10$, and $p+14$ is prime.

![](https://cdn.mathpix.com/cropped/2024_05_21_00604dc020e3721fc1f4g-044.jpg?height=859&width=1011&top_left_y=1255&top_left_x=454)
20. All natural numbers can be divided into 3 groups:
1) divisible by 3, i.e., of the form $3 k$;
2) giving a remainder of 1 when divided by 3, i.e., of the form $3 k+1$;
3) giving a remainder of 2 when divided by 3, i.e., of the form $3 k+2$.

If $p=3 k$, then it is prime only when $k=1$; then $p+10=13$ and $p+14=17$. If $p=3 k+1$, then $p+14=3 k+15=3(k+5)$-composite. If $p=3 k+2$, then $p+10=3 k+12=3(k+4)$-composite. We have the solution: $p=3$.