PROBLEM LINK:
Contest Division 1
Contest Division 2
Contest Division 3
Contest Division 4
Setter: Satyam
Tester: Abhinav Sharma, Nishank Suresh
Editorialist: Pratiyush Mishra
DIFFICULTY:
2157
PREREQUISITES:
None
PROBLEM:
You are given an integer N.
You are asked to find total number of integer pairs (A,B) such that
- 1 \leq A,B \leq N
- A^2+B^2+gcd^2(A,B)+lcm^2(A,B)=N. Note that gcd^2(A, B) and lcm^2(A, B) denote the square of gcd and the square of lcm of numbers A and B respectively.
EXPLANATION:
GCD Property: For two numbers A and B, we have:
Here we can loop A from 1 to \lfloor \sqrt{N} \rfloor, and for each value of A we can loop the LCM from A to \lfloor \sqrt{N} \rfloor with increment of A, since LCM would always be a multiple of A. This would cover all the possible values of A and lcm(A,B) and from this we can calculate the value of B as follows:
Now that we got a value of B^2, we first check if its a positive perfect square or not. If not we move to next iteration otherwise we calculate the value of B and check for two conditions:
- lcm(A,B) = LCM
- A^2 + B^2 + gcd^2(A,B) + lcm^2(A,B) = N
If both of these conditions satisfy, this means we found a pair (A,B) that satisfies the above equation. Like this we keep going to calculate the count of all possible pairs that satisfies this equation.
TIME COMPLEXITY:
O(\sqrt{N}log\sqrt{N}), for each test case.