Smallest integer > 1 which divides every element of the given array: Using C++

In this article, we are given integers in an array, and we must find the smallest number which is greater than 1 that divides all the elements in the array. For example, let us consider a sample array [30, 90, 15, 45, 165].

vector<int> arr = {30, 90, 15, 45, 165};
result = solve(arr);

Now we can find the array's GCD(greatest common divisor). If it comes out to be 1 that means that only 1 can divide the whole array, and we can return -1 or "Not possible." If it's an integer, then this integer divides the whole array. Still, this integer might not be the smallest integer that divides the whole array. Interestingly, the factors of this integer also divide the complete array, which makes sense. So if we can find the minimum factor of this integer(GCD), we get our smallest integer which divides the whole array. So, in short, we need to find the GCD of the array, and then the minimum factor of the GCD is our answer.

Example

The following C++ code finds a smallest integer greater than 1 that divides all the elements of an array. This can be done by finding the GCD for the list of elements ?

<div class="execute"></div><div class="code-mirror  language-cpp" contenteditable="plaintext-only" spellcheck="false" style="outline: none; overflow-wrap: break-word; overflow-y: auto; white-space: pre-wrap;"><span class="token macro property"><span class="token directive-hash">#</span><span class="token directive keyword">include</span> <span class="token string"><iostream></span></span>
<span class="token macro property"><span class="token directive-hash">#</span><span class="token directive keyword">include</span> <span class="token string"><vector></span></span>
<span class="token macro property"><span class="token directive-hash">#</span><span class="token directive keyword">include</span> <span class="token string"><algorithm></span></span>
<span class="token keyword">using</span> <span class="token keyword">namespace</span> std<span class="token punctuation">;</span>
<span class="token keyword">int</span> <span class="token function">divisor</span><span class="token punctuation">(</span><span class="token keyword">int</span> x<span class="token punctuation">)</span> <span class="token punctuation">{</span>
   <span class="token keyword">if</span> <span class="token punctuation">(</span>x<span class="token operator">%</span><span class="token number">2</span> <span class="token operator">==</span> <span class="token number">0</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
      <span class="token keyword">return</span> <span class="token number">2</span><span class="token punctuation">;</span>
   <span class="token punctuation">}</span>
   <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i<span class="token operator">=</span><span class="token number">3</span><span class="token punctuation">;</span>i<span class="token operator">*</span>i<span class="token operator"><=</span>x<span class="token punctuation">;</span>i<span class="token operator">+=</span><span class="token number">2</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
      <span class="token keyword">if</span> <span class="token punctuation">(</span>x<span class="token operator">%</span>i <span class="token operator">==</span> <span class="token number">0</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
         <span class="token keyword">return</span> i<span class="token punctuation">;</span>
      <span class="token punctuation">}</span>
   <span class="token punctuation">}</span>
   <span class="token keyword">return</span> x<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">int</span> <span class="token function">solve</span><span class="token punctuation">(</span>vector<span class="token operator"><</span><span class="token keyword">int</span><span class="token operator">></span> arr<span class="token punctuation">)</span> <span class="token punctuation">{</span>
   <span class="token keyword">int</span> gcd <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
   <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span>i<span class="token operator"><</span>arr<span class="token punctuation">.</span><span class="token function">size</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
      gcd <span class="token operator">=</span> <span class="token function">__gcd</span><span class="token punctuation">(</span>gcd<span class="token punctuation">,</span> arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
   <span class="token punctuation">}</span>
   <span class="token keyword">return</span> <span class="token function">divisor</span><span class="token punctuation">(</span>gcd<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">int</span> <span class="token function">main</span><span class="token punctuation">(</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
   vector<span class="token operator"><</span><span class="token keyword">int</span><span class="token operator">></span> arr <span class="token operator">=</span> <span class="token punctuation">{</span><span class="token number">30</span><span class="token punctuation">,</span> <span class="token number">90</span><span class="token punctuation">,</span> <span class="token number">15</span><span class="token punctuation">,</span> <span class="token number">45</span><span class="token punctuation">,</span> <span class="token number">165</span><span class="token punctuation">}</span><span class="token punctuation">;</span>
   cout <span class="token operator"><<</span> <span class="token function">solve</span><span class="token punctuation">(</span>arr<span class="token punctuation">)</span><span class="token punctuation">;</span>
   <span class="token keyword">return</span> <span class="token number">0</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
</div><div class="output-wrapper"><div class="console-close"></div><div class="code-output"></div></div>

Output

3

Example

If there are a lot of queries, finding the prime factor for numbers will be repetitive. Using a sieve, we can calculate the prime factors for numbers.

Another implementation in C++ for finding the smallest number greater than 1, is as follows ?

<div class="execute"></div><div class="code-mirror  language-cpp" contenteditable="plaintext-only" spellcheck="false" style="outline: none; overflow-wrap: break-word; overflow-y: auto; white-space: pre-wrap;"><span class="token macro property"><span class="token directive-hash">#</span><span class="token directive keyword">include</span> <span class="token string"><iostream></span></span>
<span class="token macro property"><span class="token directive-hash">#</span><span class="token directive keyword">include</span> <span class="token string"><vector></span></span>
<span class="token macro property"><span class="token directive-hash">#</span><span class="token directive keyword">include</span> <span class="token string"><algorithm></span></span>
<span class="token keyword">using</span> <span class="token keyword">namespace</span> std<span class="token punctuation">;</span>
<span class="token keyword">const</span> <span class="token keyword">int</span> MAX <span class="token operator">=</span> <span class="token number">100005</span><span class="token punctuation">;</span>
vector<span class="token operator"><</span><span class="token keyword">int</span><span class="token operator">></span> <span class="token function">prime</span><span class="token punctuation">(</span>MAX<span class="token punctuation">,</span> <span class="token number">0</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">void</span> <span class="token function">sieve</span><span class="token punctuation">(</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
   prime<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span> <span class="token operator">=</span> <span class="token number">1</span><span class="token punctuation">;</span>
   prime<span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">=</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">;</span>
   <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i<span class="token operator">=</span><span class="token number">2</span><span class="token punctuation">;</span> i<span class="token operator">*</span>i<span class="token operator"><</span>MAX<span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
      <span class="token keyword">if</span> <span class="token punctuation">(</span>prime<span class="token punctuation">[</span>i<span class="token punctuation">]</span> <span class="token operator">==</span> <span class="token number">0</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
         <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> j<span class="token operator">=</span>i<span class="token operator">*</span><span class="token number">2</span><span class="token punctuation">;</span>j<span class="token operator"><</span>MAX<span class="token punctuation">;</span>j<span class="token operator">+=</span>i<span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">if</span> <span class="token punctuation">(</span>prime<span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">==</span> <span class="token number">0</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
               prime<span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">=</span> i<span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
         <span class="token punctuation">}</span>
      <span class="token punctuation">}</span>
   <span class="token punctuation">}</span>
   <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i<span class="token operator">=</span><span class="token number">2</span><span class="token punctuation">;</span> i<span class="token operator"><</span>MAX<span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
      <span class="token keyword">if</span> <span class="token punctuation">(</span><span class="token operator">!</span>prime<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
         prime<span class="token punctuation">[</span>i<span class="token punctuation">]</span> <span class="token operator">=</span> i<span class="token punctuation">;</span>
      <span class="token punctuation">}</span>
   <span class="token punctuation">}</span>
<span class="token punctuation">}</span>
<span class="token keyword">int</span> <span class="token function">solve</span><span class="token punctuation">(</span>vector<span class="token operator"><</span><span class="token keyword">int</span><span class="token operator">></span> arr<span class="token punctuation">)</span> <span class="token punctuation">{</span>
   <span class="token keyword">int</span> gcd <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
   <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span> i<span class="token operator"><</span>arr<span class="token punctuation">.</span><span class="token function">size</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
      gcd <span class="token operator">=</span> <span class="token function">__gcd</span><span class="token punctuation">(</span>gcd<span class="token punctuation">,</span> arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
   <span class="token punctuation">}</span>
   <span class="token keyword">return</span> prime<span class="token punctuation">[</span>gcd<span class="token punctuation">]</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">int</span> <span class="token function">main</span><span class="token punctuation">(</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
   <span class="token function">sieve</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
   vector<span class="token operator"><</span><span class="token keyword">int</span><span class="token operator">></span> arr <span class="token operator">=</span> <span class="token punctuation">{</span> <span class="token number">30</span><span class="token punctuation">,</span> <span class="token number">90</span><span class="token punctuation">,</span> <span class="token number">15</span><span class="token punctuation">,</span> <span class="token number">45</span><span class="token punctuation">,</span> <span class="token number">165</span> <span class="token punctuation">}</span><span class="token punctuation">;</span>
   cout <span class="token operator"><<</span> <span class="token function">solve</span><span class="token punctuation">(</span>arr<span class="token punctuation">)</span><span class="token punctuation">;</span>
   <span class="token keyword">return</span> <span class="token number">0</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
</div><div class="output-wrapper"><div class="console-close"></div><div class="code-output"></div></div>

Output

3

Conclusion

We have used the sqrt(n) method to get the minimum factor. This can be optimized, and I leave that for you to try. The time complexity is O(sqrt(n)). In the second approach, the time complexity will be of the sieve, which is O(nlog(log(n))). Note that we can find the sieve up to a limit, which we set through the MAX global variable.