Modular Exponentiation 23^1002 mod 41

Enter Modular Exponentiation

Solve 231002 mod 41 using:

Modular exponentiation

Build an algorithm:

n is our exponent = 1002

y = 1 and u ≡ 23 mod 41 = 23

See here

n = 1002 is even

Since 1002 is even, we keep y = 1

Determine u2 mod p

u2 mod p = 232 mod 41

u2 mod p = 529 mod 41

529 mod 41 = 37
Reset u to this value

Cut n in half and take the integer

1002 ÷ 2 = 501

n = 501 is odd

Since 501 is odd, calculate (y)(u) mod p

(y)(u) mod p = (1)(37) mod 41

(y)(u) mod p = 37 mod 41

37 mod 41 = 37
Reset y to this value

Determine u2 mod p

u2 mod p = 372 mod 41

u2 mod p = 1369 mod 41

1369 mod 41 = 16
Reset u to this value

Cut n in half and take the integer

501 ÷ 2 = 250

n = 250 is even

Since 250 is even, we keep y = 37

Determine u2 mod p

u2 mod p = 162 mod 41

u2 mod p = 256 mod 41

256 mod 41 = 10
Reset u to this value

Cut n in half and take the integer

250 ÷ 2 = 125

n = 125 is odd

Since 125 is odd, calculate (y)(u) mod p

(y)(u) mod p = (37)(10) mod 41

(y)(u) mod p = 370 mod 41

370 mod 41 = 1
Reset y to this value

Determine u2 mod p

u2 mod p = 102 mod 41

u2 mod p = 100 mod 41

100 mod 41 = 18
Reset u to this value

Cut n in half and take the integer

125 ÷ 2 = 62

n = 62 is even

Since 62 is even, we keep y = 1

Determine u2 mod p

u2 mod p = 182 mod 41

u2 mod p = 324 mod 41

324 mod 41 = 37
Reset u to this value

Cut n in half and take the integer

62 ÷ 2 = 31

n = 31 is odd

Since 31 is odd, calculate (y)(u) mod p

(y)(u) mod p = (1)(37) mod 41

(y)(u) mod p = 37 mod 41

37 mod 41 = 37
Reset y to this value

Determine u2 mod p

u2 mod p = 372 mod 41

u2 mod p = 1369 mod 41

1369 mod 41 = 16
Reset u to this value

Cut n in half and take the integer

31 ÷ 2 = 15

n = 15 is odd

Since 15 is odd, calculate (y)(u) mod p

(y)(u) mod p = (37)(16) mod 41

(y)(u) mod p = 592 mod 41

592 mod 41 = 18
Reset y to this value

Determine u2 mod p

u2 mod p = 162 mod 41

u2 mod p = 256 mod 41

256 mod 41 = 10
Reset u to this value

Cut n in half and take the integer

15 ÷ 2 = 7

n = 7 is odd

Since 7 is odd, calculate (y)(u) mod p

(y)(u) mod p = (18)(10) mod 41

(y)(u) mod p = 180 mod 41

180 mod 41 = 16
Reset y to this value

Determine u2 mod p

u2 mod p = 102 mod 41

u2 mod p = 100 mod 41

100 mod 41 = 18
Reset u to this value

Cut n in half and take the integer

7 ÷ 2 = 3

n = 3 is odd

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

(y)(u) mod p = (16)(18) mod 41

(y)(u) mod p = 288 mod 41

288 mod 41 = 1
Reset y to this value

Determine u2 mod p

u2 mod p = 182 mod 41

u2 mod p = 324 mod 41

324 mod 41 = 37
Reset u to this value

Cut n in half and take the integer

3 ÷ 2 = 1

n = 1 is odd

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

(y)(u) mod p = (1)(37) mod 41

(y)(u) mod p = 37 mod 41

37 mod 41 = 37
Reset y to this value

Determine u2 mod p

u2 mod p = 372 mod 41

u2 mod p = 1369 mod 41

1369 mod 41 = 16
Reset u to this value

Cut n in half and take the integer

1 ÷ 2 = 0

Because n = 0, we stop

We have our answer

Final Answer

231002 mod 41 ≡ 37

What is the Answer?

231002 mod 41 ≡ 37

How does the Modular Exponentiation and Successive Squaring Calculator work?

Free Modular Exponentiation and Successive Squaring Calculator - Solves xn mod p using the following methods:
* Modular Exponentiation
* Successive Squaring
This calculator has 1 input.

What 1 formula is used for the Modular Exponentiation and Successive Squaring Calculator?

Successive Squaring I = number of digits in binary form of n. Run this many loops of a2 mod p

For more math formulas, check out our Formula Dossier

What 6 concepts are covered in the Modular Exponentiation and Successive Squaring Calculator?

exponentThe power to raise a numberintegera whole number; a number that is not a fraction
...,-5,-4,-3,-2,-1,0,1,2,3,4,5,...modular exponentiationthe remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus)modulusthe remainder of a division, after one number is divided by another.
a mod bremainderThe portion of a division operation leftover after dividing two integerssuccessive squaringan algorithm to compute in a finite field