Problem 9.4. It is known that the number 400000001 is the product of two prime numbers $p$ and $q$. Find the sum of the natural divisors of the number $p+q-1$.

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Answer: 45864.

Solution. The number $n=400000001$ can be written in the following form

\[
\begin{aligned}
n & =4 \cdot 10^{8}+1=4 \cdot 10^{8}+4 \cdot 10^{4}+1-4 \cdot 10^{4}= \\
& =\left(2 \cdot 10^{4}+1\right)^{2}-\left(2 \cdot 10^{2}\right)^{2}=\left(2 \cdot 10^{4}+2 \cdot 10^{2}+1\right)\left(2 \cdot 10^{4}-2 \cdot 10^{2}+1\right)
\end{aligned}
\]

Since $n$ is the product of two prime numbers $p$ and $q$, the factorization obtained above is exactly this. Therefore,

\[
p+q-1=2 \cdot 10^{4}+2 \cdot 10^{2}+1+2 \cdot 10^{4}-2 \cdot 10^{2}+1-1=4 \cdot 10^{4}+1
\]

To factorize the last number, we can use the same trick.

\[
\begin{aligned}
4 \cdot 10^{4}+1 & =4 \cdot 10^{4}+4 \cdot 10^{2}+1-4 \cdot 10^{2}=\left(2 \cdot 10^{2}+1\right)^{2}-(2 \cdot 10)^{2}= \\
& =(201-20)(201+20)=181 \cdot 221=13 \cdot 17 \cdot 181
\end{aligned}
\]

The sum of the natural divisors of the obtained number can be found, for example, using the formula

\[
(13+1)(17+1)(181+1)=45864
\]

## Criteria

The highest appropriate criterion is used:

76. Any correct solution to the problem.

5 6. The correct prime factorization of the number $p+q-1$ is obtained, but due to an error, an incorrect answer is given.

4 6. The answer is given under the assumption that $p+q-1=181 \cdot 221$ is the prime factorization.

3 6. The value of $p+q-1$ is correctly calculated.

In any of the above cases, points are deducted for the following deficiency:

-2 6. The correct factorization of the given number in the problem is provided, but it is not shown where this factorization comes from, nor that it indeed gives the required number when multiplied.