Long Products

Define $F(m,n)$ as the number of $n$-tuples of positive integers for which the product of the elements doesn’t exceed $m$.
$F(10, 10) = 571$.
$F(10^6, 10^6) \bmod 1\,234\,567\,891 = 252903833$.
Find $F(10^9, 10^9) \bmod 1\,234\,567\,891$.

The problem involves two concepts: multiplicative number theory and combinatorics. Let’s start by understanding the function $F(m, n)$.

$F(m, n)$ represents the number of $n$-tuples for which the product of the elements does not exceed $m$. This problem thus asks how many combinations of $n$ numbers can be multiplied together without the product exceeding $m$.

This seems a complex problem with $10^9$ tuples and a product limit of $10^9$. However, it’s much more manageable if we understand the mathematical principles behind it.

We can see $F(m, n)$ as the number of ways you can distribute $n$ factors among the prime factorization of $m$.

In this problem, the given modulus, $1234567891$, is a large prime number, which could be a hint to use the Fermat’s Little Theorem.

Before we go into evaluating the expression, let’s understand how we can calculate $F(m, n)$.

For a large $m$, we can write all factors in the form of prime powers $p^i$. Essentially, the problem is to distribute $n$ numbers to these powers such that the sum does not exceed $m$.

This problem is analogous to a combinatorial concept known as “stars and bars”, where we distribute $n$ identical things among $r$ groups. The number of ways to do it is given by $\binom{r+n-1}{n}$, where $\binom{n}{r}$ denotes “n choose r”, or the number of ways to choose $r$ items from $n$ without order mattering.

In the same way, with $F(m, n)$, we distribute $n$ over the “groups” of the powers of each prime factor in $m$.

However to conclude, this problem seems to be a problem from International Mathematical Olympiad which is highly nontrivial. Solving it efficiently, particularly for large values as requested, requires high-level techniques and deep understanding of number theory and combinatorics.

To solve the exact case of $F(10^9, 10^9) \bmod 1234567891$, without access to relevant high-speed computing resources and mathematical software, may not be practically feasible due to the enormity of the numbers involved. It would necessitate an ingenious approach or insight not easily obtained.

In general, problems of this type, with specific large numbers, are often beyond the scope of hand calculation and need high speed computation to solve.

Sorry for not providing you exact answer to this, but due to the enormity of numbers and complexities involved this may need a good computation resource.

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