I have stumbled upon a task to calculate modulo of 5 * (2¹⁰)¹⁸ where mod = 10⁹+7

Using multiplication formula

x * y % mod = ((x % mod) * (y % mod)) % mod

we can first split our problem to

(n % mod) * ((2 ¹⁰)¹⁸ % mod) % mod = X

and then to

(n % mod) * ((2 ¹⁰*¹⁸ % mod) % mod = X

Modern programing languages like python can do this already in pow (exponent) function has third parameter for mod.

(5 * pow(2, 10 * 18, mod) ) % mod = 256609439 Interesting enough python pow support this but math.pow does not 😬

Behind the scenes language can do this in log(n) complexity using same multiplication principle as described above. Lets implement this ourselves in case your language does not support this like golang.

package main

import "fmt"

const mod int64 = 1e9 + 7

func main() {
	fmt.Printf("%d", (5 * pow(2, 10*18) % mod))
}

func pow(base, exp int64) int64 {
	var res int64 = 1
	for exp > 0 {
		if exp&1 == 1 {
			res *= base
			res %= mod
		}
		base *= base
		base %= mod
		exp /= 2
	}
	return res
}