Given integers x1, x2, x3……xn, b and m, we are supposed to find the result of ((x1*x2….xn)/b)mod(m).
Example 1 : Suppose that we are required to find (55C5)%(1000000007) i.e ((55*54*53*52*51)/120)%1000000007
Naive Method :
- Simply calculate the product (55*54*53*52*51)= say x,
- Divide x by 120 and then take its modulus with 1000000007
Using Modular Multiplicative Inverse :
Above method will work only when x1, x2, x3….xn have small values.
Suppose we are required to find the result where x1, x2, ….xn fall in the range of ~1000000(10^6). So we will have to exploit the rule of modular mathematics which says :
Note that the above formula is valid for modular multiplication. Similar formula for division does not exist.
i.e (a/b)mod(m) != a(mod(m))/b(mod(m))
- So we are required to find out modular multiplicative inverse of b say i and then multiply ‘i’ with a .
- After this we will have to take the modulus of the result obtained.
i.e ((x1*x2….xn)/b)mod(m)=((x1*x2….xn)*i)mod(m)= ((x1)mod(m) * (x2)mod(m) *…. (xn)mod(m) * (i)mod(m))mod(m)
Note : To find modular multiplicative inverse we can use Extended Eucledian algorithm or Fermat’s Little Theorem.
Example 2 : Let us suppose that we have to find (55555C5)%(1000000007) i.e ((55555*55554*55553*55552*55551)/120)%1000000007.
Input : Output :Answer using naive method: -5973653 Answer using multiplicative modular inverse concept: 300820513
It is clear from the above example that the naive method will lead to overflow of data resulting in incorrect answer. Moreover, using modular inverse will give us the correct answer.
Without Using Modular Multiplicative Inverse :
But it is interesting to note that a slight change in code will discard the use of finding modular multiplicative inverse.
Answer using shortcut: 300820513
Why did it work?
Let us consider a = x1*x2*x3…….xn
We have to find ans = (a/b)%1000000007
- Let result of a%(1000000007*b) be y. To avoid overflow, we use modular multiplicative property. This can be represented as
a = (1000000007*b)q + y where y < (1000000007*b) and q is an integer
- Now dividing LHS and RHS by b, we get
y/b = a/b -(1000000007*b)*q/b
= a/b -1000000007*q < 1000000007 (From 1)
Therefore, y/b is equivalent to (a/b)mod(b*1000000007). 🙂
- Modular Division
- Modular Exponentiation (Power in Modular Arithmetic)
- Modular exponentiation (Recursive)
- Modular multiplicative inverse from 1 to n
- Modular multiplicative inverse
- Number of solutions to Modular Equations
- How to avoid overflow in modular multiplication?
- Find modular node in a linked list
- Using Chinese Remainder Theorem to Combine Modular equations
- DFA based division
- Division without using '/' operator
- Find the number after successive division
- Cyclic Redundancy Check and Modulo-2 Division
- Write you own Power without using multiplication(*) and division(/) operators
- Number of digits before the decimal point in the division of two numbers
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.