Animated Solution for Mathematics - Permutations and Combinations: The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.

To find the total number of four-digit integers $abcd$ where each of the first three digits ($a,b,c$) is divisible by the last digit ($d$), we must first define our constraints. The digits $a,b,c,d$ are elements of the set $\{0,1,2,\dots ,9\}$.

The leading digit $a$ cannot be $0$, as it must be a four-digit number. Furthermore, since $d$ acts as a divisor, $d$ cannot be $0$. Thus, our master variable $d$ is restricted to the set $\{1,2,\dots ,9\}$.

Divide and Conquer

Because the set of multiples of $d$ changes depending on the value of $d$, we must treat each value as a unique case. We will calculate the number of valid combinations for each $d$ and then aggregate the results.

For $d=1$, every digit is divisible by $1$. The digit $a$ has $9$ options $\{1,\dots ,9\}$, while $b$ and $c$ each have $10$ options $\{0,\dots ,9\}$.

For $d=2$, the digits must be even. The digit $a$ has $4$ options $\{2,4,6,8\}$, while $b$ and $c$ each have $5$ options $\{0,2,4,6,8\}$.

For $d=3$, the multiples are $\{3,6,9\}$ for $a$ ($3$ options) and $\{0,3,6,9\}$ for $b$ and $c$ ($4$ options each).

For $d=4$, the multiples are $\{4,8\}$ for $a$ ($2$ options) and $\{0,4,8\}$ for $b$ and $c$ ($3$ options each).

For $d\in \{5,6,7,8,9\}$, the only non-zero multiple of $d$ less than $10$ is $d$ itself. Therefore, $a$ has only $1$ option (the digit $d$).

For $b$ and $c$, the valid choices are $\{0,d\}$, providing $2$ options each. For each of these five values of $d$, the number of ways is:

Since there are $5$ such values of $d$, the total for this range is:

We now sum the results from all individual cases to find the total count of such four-digit integers:

The total number of such four-digit integers is $1086$.