The number 3798125P369 is divisible by 7. What is the value of the digit P?

The correct option is 6.

We are given the number 3798125P369, which is divisible by 7, and we need to find the single digit value of P from the given options.

To determine the divisibility of a large number by 7, we can use modular arithmetic. A number is divisible by 7 if the number modulo 7 is equal to 0.

Let's break the given 11-digit number 3798125P369 into expanded positional components by separating the digit P:

$3798125\text{P}369=37981250000+\text{P}\times {10}^3+369$

Now, let's evaluate each component modulo 7:

First, find the remainder when 37981250000 is divided by 7:

$37981250000÷7=5425892857\text{with a remainder of}1$

So,
$37981250000\equiv 1\left(\text{mod}7\right)$

Next, find the remainder of the coefficient of P, which is 10³ = 1000, modulo 7:

$1000÷7=142\text{with a remainder of}6$

So,
$1000\equiv 6\equiv -1\left(\text{mod}7\right)$

Next, find the remainder when 369 is divided by 7:

$369÷7=52\text{with a remainder of}5$

So,
$369\equiv 5\left(\text{mod}7\right)$

Now, combine these modulo 7 equivalences into the total expression for the number:

$3798125\text{P}369\equiv 1+6\text{P}+5\left(\text{mod}7\right)$

$3798125\text{P}369\equiv 6\text{P}+6\left(\text{mod}7\right)$

For the number to be divisible by 7, the total remainder must be 0 modulo 7:

$6\text{P}+6\equiv 0\left(\text{mod}7\right)$

We can factor out 6:

$6(\text{P}+1)\equiv 0\left(\text{mod}7\right)$

Since 6 and 7 are coprime, this implies:

$\text{P}+1\equiv 0\left(\text{mod}7\right)$

$\text{P}\equiv 6\left(\text{mod}7\right)$

Since P is a single digit (from 0 to 9), the only possible value for P is 6.