Solve 11^{13} mod 53 using:

Modular exponentiation

n is our exponent = 13

y = 1 and u ≡ 11 mod 53 = 11

See here

Since 13 is odd, calculate (y)(u) mod p

(y)(u) mod p = (1)(11) mod 53

(y)(u) mod p = 11 mod 53

11 mod 53 = 11

Reset y to this value

u^{2} mod p = 11^{2} mod 53

u^{2} mod p = 121 mod 53

121 mod 53 = 15

Reset u to this value

13 ÷ 2 = 6

Since 6 is even, we keep y = 11

u^{2} mod p = 15^{2} mod 53

u^{2} mod p = 225 mod 53

225 mod 53 = 13

Reset u to this value

6 ÷ 2 = 3

Since 3 is odd, calculate (y)(u) mod p

(y)(u) mod p = (11)(13) mod 53

(y)(u) mod p = 143 mod 53

143 mod 53 = 37

Reset y to this value

u^{2} mod p = 13^{2} mod 53

u^{2} mod p = 169 mod 53

169 mod 53 = 10

Reset u to this value

3 ÷ 2 = 1

Since 1 is odd, calculate (y)(u) mod p

(y)(u) mod p = (37)(10) mod 53

(y)(u) mod p = 370 mod 53

370 mod 53 = 52

Reset y to this value

u^{2} mod p = 10^{2} mod 53

u^{2} mod p = 100 mod 53

100 mod 53 = 47

Reset u to this value

1 ÷ 2 = 0

We have our answer

11^{13} mod 53 ≡ **52**

Solve 11^{13} mod 53 using:

the Successive Squaring Method

Using our binary calculator, we see that 0 in binary form is

The length of this binary term is 0, so this is how many steps we will take for our algorithm below

i | a | a^{2} | a^{2} mod p |
---|

Look at the binary term with values of 1 in red

This signifies which terms we use for expansion:

= 1 mod 53 = **1**

11^{13} mod 53 ≡ **52**

= 1 mod 53 =**1**

= 1 mod 53 =

11^{13} mod 53 ≡ **52**

= 1 mod 53 =**1**

= 1 mod 53 =

Free Modular Exponentiation and Successive Squaring Calculator - Solves x^{n} mod p using the following methods:

* Modular Exponentiation

* Successive Squaring

This calculator has 1 input.

* Modular Exponentiation

* Successive Squaring

This calculator has 1 input.

Successive Squaring I = number of digits in binary form of n. Run this many loops of a^{2} mod p

For more math formulas, check out our Formula Dossier

For more math formulas, check out our Formula Dossier

- exponent
- The power to raise a number
- integer
- a whole number; a number that is not a fraction

...,-5,-4,-3,-2,-1,0,1,2,3,4,5,... - modular exponentiation
- the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus)
- modulus
- the remainder of a division, after one number is divided by another.

a mod b - remainder
- The portion of a division operation leftover after dividing two integers
- successive squaring
- an algorithm to compute in a finite field

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