On the existence of latin squares with special distribution properties

An n×nn×n latin square (ar,c|0≤r,c≤n−1)(ar,c|0≤r,c≤n−1) is an (s,t)(s,t) latin square if every subrectangle Ri,jRi,j, 0≤i,j≤n−10≤i,j≤n−1, consisting of cells {ai+k,j+ℓ|0≤kstn>st and the product of the greatest common divisors gcd(s,n)gcd(t,n)gcd(s,n)gcd(t,n) is a divisor of nn.