fix(learn): Rework Euler Problem 182 (#41337)

* Rework Euler Problem 182

* Reordered tests

* Fixes for formatting

curriculum/challenges/english/10-coding-interview-prep/project-euler/problem-182-rsa-encryption.md

@@ -10,16 +10,59 @@ dashedName: problem-182-rsa-encryption
 
 The RSA encryption is based on the following procedure:
 
-Generate two distinct primes p and q.Compute n=pq and φ=(p-1)(q-1).
+Generate two distinct primes `p` and `q`.
+Compute `n=p*q` and `φ=(p-1)(q-1)`.
+Find an integer `e`, `1 < e < φ`, such that `gcd(e,φ) = 1`
 
-Find an integer e, 1&lt;e&lt;φ, such='' that='' gcd(e,φ)='1.' a='' message='' in='' this='' system='' is='' number='' the='' interval='' \[0,n-1].='' text='' to='' be='' encrypted='' then='' somehow='' converted='' messages='' (numbers='' \[0,n-1]).='' encrypt='' text,='' for='' each='' message,='' m,='' c='me' mod='' n='' calculated.='' decrypt='' following='' procedure='' needed:='' calculate='' d='' ed='1' φ,='' c,='' m='cd' n.='' there='' exist='' values='' of='' e='' and='' me='' call='' which='' unconcealed='' messages.='' an='' issue='' when='' choosing='' should='' not='' too='' many='' instance,='' let='' p='19' q='37.' φ='18\*36=648.' if='' we='' choose='' then,='' although='' gcd(181,648)='1' it='' turns='' out='' all='' possible='' messagesm='' (0≤m≤n-1)='' are='' calculating='' any='' valid='' choice='' some='' it's='' important='' at='' minimum.='' find='' sum='' e,='' 1&lt;e&lt;φ(1009,3643)='' so='' value='' &lt;='' section=''>&lt;/e&lt;φ,>
+A message in this system is a number in the interval `[0,n-1]`.
+A text to be encrypted is then somehow converted to messages (numbers in the interval `[0,n-1]`).
+To encrypt the text, for each message, `m`, c=m<sup>e</sup> mod n is calculated.
+
+To decrypt the text, the following procedure is needed: calculate `d` such that `ed=1 mod φ`, then for each encrypted message, `c`, calculate m=c<sup>d</sup> mod n.
+
+There exist values of `e` and `m` such that m<sup>e</sup> mod n = m.
+We call messages `m` for which m<sup>e</sup> mod n=m unconcealed messages.
+
+An issue when choosing `e` is that there should not be too many unconcealed messages.
+For instance, let `p=19` and `q=37`.
+Then `n=19*37=703` and `φ=18*36=648`.
+If we choose `e=181`, then, although `gcd(181,648)=1` it turns out that all possible messages
+m `(0≤m≤n-1)` are unconcealed when calculating m<sup>e</sup> mod n.
+For any valid choice of `e` there exist some unconcealed messages.
+It's important that the number of unconcealed messages is at a minimum.
+
+For any given `p` and `q`, find the sum of all values of `e`, `1 < e < φ(p,q)` and `gcd(e,φ)=1`, so that the number of unconcealed messages for this value of `e` is at a minimum.
 
 # --hints--
 
-`euler182()` should return 399788195976.
+`RSAEncryption` should be a function.
 
 ```js
-assert.strictEqual(euler182(), 399788195976);
+assert(typeof RSAEncryption === 'function')
+```
+
+`RSAEncryption` should return a number.
+
+```js
+assert.strictEqual(typeof RSAEncryption(19, 37), 'number');
+```
+
+`RSAEncryption(19, 37)` should return `17766`.
+
+```js
+assert.strictEqual(RSAEncryption(19, 37), 17766);
+```
+
+`RSAEncryption(283, 409)` should return `466196580`.
+
+```js
+assert.strictEqual(RSAEncryption(283, 409), 466196580);
+```
+
+`RSAEncryption(1009, 3643)` should return `399788195976`.
+
+```js
+assert.strictEqual(RSAEncryption(19, 37), 17766);
 ```
 
 # --seed--
@@ -27,16 +70,44 @@ assert.strictEqual(euler182(), 399788195976);
 ## --seed-contents--
 
 ```js
-function euler182() {
+function RSAEncryption(p, q) {
 
   return true;
 }
 
-euler182();
+RSAEncryption(19, 37);
 ```
 
 # --solutions--
 
 ```js
-// solution required
+function gcd(a, b) {
+    if (b)
+        return gcd(b, a % b);
+    else
+        return a;
+}
+
+function RSAEncryption(p, q) {
+    let phi = (p - 1) * (q - 1);
+
+    let best = Number.MAX_SAFE_INTEGER;
+    let sum = 0;
+
+    for (let e = 0; e < phi; ++e) {
+        if (!(gcd(e, phi) == 1))
+            continue;
+
+        let msg = (gcd(p - 1, e - 1) + 1) * (gcd(q - 1, e - 1) + 1);
+
+        if (best == msg) {
+            sum += e;
+        } else if (best > msg) {
+            best = msg;
+            sum = e;
+        }
+    }
+
+    return sum;
+}
 ```