Use the AND and OR Operators
The code has been modified slightly and now uploaded to the site.

324.cpp
#include <iostream>
using namespace std;
int main(){
int input;
bool andDivisible = false;
bool orDivisible = false;
bool orDivisibleNotBoth = false;
std::cout << "Enter an integer: \n";
std::cin >> input;
if (input % 5 == 0 && input % 6 == 0){
andDivisible = true;
}
if (input % 5 == 0 || input % 6 == 0){
orDivisible = true;
}
if (input % 6 == 0 ^ input % 5 == 0){
orDivisibleNotBoth = true;
}
std::cout << "Is " << input << " divisible by 5 and 6? " << andDivisible << "\n";
std::cout << "Is " << input << " divislbe by 5 or 6? " << orDivisible << "\n";
std::cout << "Is " << input << " divisible by 5 or 6, but not both? " << orDivisibleNotBoth << "\n";
}