I have some natural numbers whose only common divisor is 1. If I add 50 to each of them, I find two-digit numbers. If I subtract 32 from each of them, I also find two-digit natural numbers. What are the numbers that I have?
Adding $50+50=100$ we get three digits and $41-32=9$ has one, therefore, the numbers sought cannot be greater than 49, nor less than 42. Since 43 is prime (as well as 47), my numbers are any of 42, 43, $44, 45, 46, 47, 48$ and 49, which do not have a common divisor other than 1.