Find the remainder when 3^19 is divided by 19 . (1) 3 (2) 15 (3) 16 (4) None of these

Name: Problem 30, Chapter 17: The IIT Foundation series of Mathematics Class
Uploaded: 2021-09-01T07:41:22-08:00
Duration: 3 min 45 s
Channel: Kumareshwaran Rathinasabapathy
Description: Find the remainder when 3^19 is divided by 19 . (1) 3 (2) 15 (3) 16 (4) None of these

Find the remainder when 3^19 is divided by 19 . (1) 3 (2) 15...

Key Concepts

Modular Arithmetic

Modular arithmetic involves performing calculations with numbers where we consider only the remainder when one integer is divided by another integer known as the modulus. It forms the basis of congruences in number theory and has applications in solving problems involving remainders, cyclic patterns, and periodic phenomena.

Modular Exponentiation

Modular exponentiation is the process of finding the remainder when an exponential expression is divided by a modulus. It is particularly useful for efficiently computing large powers in modular arithmetic by reducing intermediate results modulo the number, thereby simplifying calculations and preventing the numbers from becoming unmanageably large.

Fermat's Little Theorem

Fermat's Little Theorem is a principle in number theory that states if a number is prime and does not divide the base, then raising the base to one less than the prime gives a result that is congruent to 1 modulo the prime. This theorem is a powerful tool in computational number theory and helps simplify calculations in modular arithmetic, especially for reducing large exponents.

Transcript

00:01 In this question, we'll find the remainder when 3 power 19 is divided by 19.

00:05 So let's see how to do this one.

00:07 First, what we can do is we can write down this 3 power 19 as 3 raised to the power of 4 and once again raised to the power of 4 multiplied by 3 raised to the power of 3.

00:20 So we can verify this.

00:21 This is 3 raised to the power of 4, this is equal to.

00:26 These two will get multiplied according to the law of exponents that is four times four is 16 and then added with this power that is 3 so basically we get the power as 19 so these two sides are equal now let's simplify this further so we have to find this 3 power 4 3 is basically 3 multiply with itself for about 4 times so we'll be getting this is 9 and this is 9 so 9 times 9 so 9 times 9 is 81 so i can replace 81 over here raise to the power of 4 similarly i'm going to i'm going to write this 3 power 3 as 3 multiplied with itself by 3 times this will be 27 so we now have this 81 raised to the power of 4 we will mod this out now with 19 which means i can write down here this 81 i have to mod it out with 19 by dividing by 19 so when i do that i'll be getting four times is 76 and then the remainder is 5 so instead of 81 i can put this as 5 after modding out with 19 so this is equal to 5 raise to the power of 4 similarly i can mod this 27 with 19 so when i do that 27 mod it out with 19 this will go only one times so then the remainder is 8 so instead of 27 i can write down this as 8 and let's simplify this further so this 5 power 4 can be written as 5 squared raised to the power of 2 and then with 8 so now we can see that 5 squared is 25 and once again raised to the power of 2 multiplied by 8 so 25 we have to mod it out with 19 so when we do that we take the 25 divided by 19 it goes only one time and the remainder is 6 so therefore this second replace it as 6 squared times 8 let's simplify this further 6 squared is equivalent to 36 and then we have 8 so now we mod out this 36 with 19 because it is greater than 19 so 19 this will go uh one times this will be we'll get the remainder as 17 so this is equivalent to 17 times 8 so we have to multiply this 17 times 8 and this comes out to be 130 so finally we just have to mod out this 136 with 19.

03:04 So we'll do it here.

03:06 This will go about seven times...

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