The largest two-digit number that satisfies 5 x ≡6(4) is (1) 96 (2) 97 (3) 98 (4) 99

Name: Problem 13, Chapter 17: The IIT Foundation series of Mathematics Class
Uploaded: 2021-08-31T16:32:48-08:00
Duration: 4 min 27 s
Channel: Kumareshwaran Rathinasabapathy
Description: The largest two-digit number that satisfies 5 x ≡6(4) is (1) 96 (2) 97 (3) 98 (4) 99

The largest two-digit number that satisfies 5 x ≡6(4) is (1)...

Key Concepts

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers 'wrap around' after reaching a certain value called the modulus. It is crucial in solving problems involving congruence relations, and it simplifies many calculations by reducing numbers modulo the chosen base.

Congruence Relations

A congruence relation specifies that two numbers have the same remainder when divided by a given modulus. This concept is fundamental in number theory, allowing us to replace numbers with their residues to simplify equations and solve problems within a cyclical number system.

Solving Linear Congruences

Solving linear congruences involves finding all integers that satisfy an equation given in modular form. This typically requires reducing the coefficients modulo the given modulus and determining the residue class that forms the solution set, which is an essential technique in modular arithmetic.

Transcript

00:01 In this question, we are given this linear congruence, 5x congruent to 6 mod 4, and we just have to determine the largest 2 digit number that must satisfy this congruence.

00:13 So let's try to figure it out.

00:15 So first, let's write down the equivalent of this congruents.

00:19 We can write down this as 5x minus 6 is equal to 4k, but k is a sum number, which means after we plug in one of the numbers given, over here that is 96 or 97 or 98 or 99 it must satisfy this equation that is after we simplify this it should be a multiple of 4 only then we can say that that particular number will satisfy this congruance so let's start off with the first number that is 96 we plug it over here 5 times of 96 minus 6 this has to be 4k that is it should be a multiple of 4 so instead of 4 so instead of doing the normal way of multiplying this 96 with 5 and then subtracting and then checking if it is a multiple of 4 what we can do is we can simply mod this 96 with 4 that is we divide this 96 with 4 and find the remainder so this will be 8 remainder is 1 i pull this 6 this will be 4 16 so i get the remainder as 0 now what i do is i take this remainder 0 and i plug it in place of 96 so this is going to be five times of 0 minus 6 let's check if it is a multiple of 4 this is going to be 5 times of 0 is 0 minus 6 so this is not a multiple of 4 so we can ignore this now let's move on to the next one that is 97 so again we are going to perform the same thing so first let's plug in the number 5 times of 97 minus 6 we have to check if it is multiple of 4.

02:09 Since we already done for 96, we can mod this 97 in the same way.

02:16 Notice that 97 is just one more than 96.

02:20 So if we mod this 97 with 4, we are going to get the remainder as 1.

02:25 So we will get something over here, something over here.

02:28 And definitely we are going to get the remainder as 1.

02:31 So you can simply take this reminder 1 and plug it over here.

02:34 This is going to be 5 times of 1 minus 6.

02:38 So this will be 5 minus 6 is minus 1.

02:41 And this is also not a multiple of 4.

02:44 So we can ignore this.

02:45 Now let's take the another number that is 98...