Analyze the Divisibility of 9P6

A number is divisible by $3$ if the sum of its digits is a multiple of $3$. For $9P6$, the sum of digits is $9 + P + 6 = 15 + P$.

$15 + P = 3m \quad (m \in \mathbb{Z})$

Find the Least Possible Value of P

Since $15$ is already divisible by $3$, the least possible non-negative single-digit value for $P$ is $0$.

$P_{\text{min}} = 0$

Solve for K

Substitute $P = 0$ into the sum $268 + 6K8 = 906$. Subtract $268$ from $906$ to find $6K8$.

$6K8 = 906 - 268 = 638 \implies K = 3$

Verify Divisibility and Least Condition

With $K = 3$ and $P = 0$, $268 + 638 = 906$, and $906$ is divisible by $3$ ($9 + 0 + 6 = 15$). This yields the least values $K = 3$ and $P = 0$.

$K = 3, \quad P = 0$