Integers in a Grid
Source: 2023 AIME II/10
February 16, 2023
AMCAIME
Problem Statement
Let be the number of ways to place the integers through in the cells of a grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by . One way to do this is shown below. Find the number of positive integer divisors of .[asy]
size(160);
defaultpen(linewidth(0.6));
for(int j=0;j<=6;j=j+1)
{
draw((j,0)--(j,2));
}
for(int i=0;i<=2;i=i+1)
{
draw((0,i)--(6,i));
}
for(int k=1;k<=12;k=k+1)
{
label("",(floor((k-1)/2)+0.5,k%2+0.5));
}
[/asy]