If $p, q, r$, and $s$ are digits, how many of the 14-digit positive integers of the form 88663311 pqr s 48 are divisible by 792 ?
(A) 48
(B) 56
(C) 40
(D) 60
(E) 50
First, we note that 88663311000 is divisible by 792 . (We can check this by division.)

Therefore, 88663311000000 is also divisible by 792 .

Since 88663311000000 is divisible by 792 , then 88663311 pqr s48 is divisible by 792 exactly when pqr s48 is divisible by 792 . (This comes from the fact that if the difference between two integers is divisible by $d$, then either both are divisible by $d$ or neither is divisible by $d$.)

The smallest integer of the form pqr s48 is 48 (which is " 000048 ") and the largest integer of the form pqr s48 is 999948 .

Since $999948 \div 792 \approx 1262.6$, then the multiples of 792 in between 48 and 999948 are the integers of the form $792 \times n$ where $1 \leq n \leq 1262$.

Suppose that $792 \times n=$ pqr s48 for some integer $n$.

Comparing units digits, we see that the units digit of $n$ must be 4 or 9 .

This means that $n=10 c+4$ or $n=10 c+9$ for some integer $c \geq 0$.

In the first case, $792(10 c+4)=7920 c+3168$.

This integer has a units digit of 8 .

For this integer to have a tens digit of 4 , we need $2 c+6$ to have a units digit of 4 , which happens exactly when $c$ has units digit 4 or 9 .

This means that $c$ can be $4,9,14,19,24, \ldots$

This means that $n$ can be $44,94,144,194,244, \ldots$

Since $1 \leq n \leq 1262$, then there are 25 possible values of $n$ with units digit 4 , because there are 2 values of $n$ between 0 and 100, 2 between 100 and 200, and so on up to 1200, with an additional 1 (namely, 1244) between 1200 and 1262.

In the second case, $792(10 c+9)=7920 c+7128$.

This integer has a units digit of 8 .

For this integer to have a tens digit of 4 , we need $2 c+2$ to have units digit 4 , which happens exactly when $c$ has units digit 1 or 6 .

This means that $c$ can be $1,6,11,16,21, \ldots$

This means that $n$ can be $19,69,119,169,219, \ldots$

Since $1 \leq n \leq 1262$, then there are again 25 possible values of $n$ with units digit 9 .

In total, there are $25+25=50$ 14-digit positive integers of the desired form that are divisible by 792 .

ANSWER: (E)