Question 51
Let us say that a pair $(a, b)$ of positive integers is nice if $a$ divides $b$ and $a+1$ divides $b+1$.
a. Prove that, for a fixed positive integer $a$, there exist infinitely many positive integers $b$ such that the pair $(a, b)$ is nice.
b. For a fixed positive integer $a$, determine the smallest integer $b\gt a$ such that $(a, b)$ is nice.
c. Now take $a=18$; determine the smallest positive integer $b\gt a$ such that $(a, b)$ is nice and, in addition, $a+2$ divides $b+2$.
a. Prove that, for a fixed positive integer $a$, there exist infinitely many positive integers $b$ such that the pair $(a, b)$ is nice.
b. For a fixed positive integer $a$, determine the smallest integer $b\gt a$ such that $(a, b)$ is nice.
c. Now take $a=18$; determine the smallest positive integer $b\gt a$ such that $(a, b)$ is nice and, in addition, $a+2$ divides $b+2$.