---
id: 5900f4c81000cf542c50ffd9
title: 'Problem 347: Largest integer divisible by two primes'
challengeType: 5
forumTopicId: 302006
dashedName: problem-347-largest-integer-divisible-by-two-primes
---

# --description--

The largest integer $≤ 100$ that is only divisible by both the primes 2 and 3 is 96, as $96 = 32 \times 3 = 2^5 \times 3$.

For two distinct primes $p$ and $q$ let $M(p, q, N)$ be the largest positive integer $≤ N$ only divisible by both $p$ and $q$ and $M(p, q, N)=0$ if such a positive integer does not exist.

E.g. $M(2, 3, 100) = 96$.

$M(3, 5, 100) = 75$ and not 90 because 90 is divisible by 2, 3 and 5. Also $M(2, 73, 100) = 0$ because there does not exist a positive integer $≤ 100$ that is divisible by both 2 and 73.

Let $S(N)$ be the sum of all distinct $M(p, q, N)$. $S(100)=2262$.

Find $S(10\\,000\\,000)$.

# --hints--

`integerDivisibleByTwoPrimes()` should return `11109800204052`.

```js
assert.strictEqual(integerDivisibleByTwoPrimes(), 11109800204052);
```

# --seed--

## --seed-contents--

```js
function integerDivisibleByTwoPrimes() {

  return true;
}

integerDivisibleByTwoPrimes();
```

# --solutions--

```js
// solution required
```