[ Divisibility of numbers. General properties ] [ Examples and counterexamples. Constructions ]

Natural numbers $A$ and $B$ are divisible by all natural numbers from 1 to 65. What is the smallest natural number that $A+B$ may not be divisible by?
From the condition, it follows that $A+B$ is divisible by all numbers from 1 to 65. This sum is also divisible by $66=2 \cdot 33$. However, $A+B$ is not necessarily divisible by 67. For example,

$A=65!, B=2 \cdot 65!, A+B=3 \cdot 65!$ which is not divisible by 67.

## Answer

Not divisible by 67.