878. Nth Magical Number

Description

A positive integer is magical if it is divisible by either a or b.

Given the three integers n, a, and b, return the nth magical number. Since the answer may be very large, return it modulo 109 + 7.

Example 1:

Input: n = 1, a = 2, b = 3
Output: 2


Example 2:

Input: n = 4, a = 2, b = 3
Output: 6


Constraints:

• 1 <= n <= 109
• 2 <= a, b <= 4 * 104

Solutions

• class Solution {
private static final int MOD = (int) 1e9 + 7;

public int nthMagicalNumber(int n, int a, int b) {
int c = a * b / gcd(a, b);
long l = 0, r = (long) (a + b) * n;
while (l < r) {
long mid = l + r >>> 1;
if (mid / a + mid / b - mid / c >= n) {
r = mid;
} else {
l = mid + 1;
}
}
return (int) (l % MOD);
}

private int gcd(int a, int b) {
return b == 0 ? a : gcd(b, a % b);
}
}

• using ll = long long;

class Solution {
public:
const int mod = 1e9 + 7;

int nthMagicalNumber(int n, int a, int b) {
int c = lcm(a, b);
ll l = 0, r = 1ll * (a + b) * n;
while (l < r) {
ll mid = l + r >> 1;
if (mid / a + mid / b - mid / c >= n)
r = mid;
else
l = mid + 1;
}
return l % mod;
}
};

• class Solution:
def nthMagicalNumber(self, n: int, a: int, b: int) -> int:
mod = 10**9 + 7
c = lcm(a, b)
r = (a + b) * n
return bisect_left(range(r), x=n, key=lambda x: x // a + x // b - x // c) % mod


• func nthMagicalNumber(n int, a int, b int) int {
c := a * b / gcd(a, b)
const mod int = 1e9 + 7
r := (a + b) * n
return sort.Search(r, func(x int) bool { return x/a+x/b-x/c >= n }) % mod
}

func gcd(a, b int) int {
if b == 0 {
return a
}
return gcd(b, a%b)
}