Question

**NUMBER THEORY**

Without calculating the products or using the calculator, find the remainders of the division. Demonstrate the process, the solution has been shown already.

a) 5²⁸⁵⁷⁴ divided by 17.

Solution : 15.

b) 35³⁴⁶ divided by 41.

Solution : 2.

c) 34 × 17 divided by 29.

Sol : 27

d) 19 × 14 divided by 23.

Sol : 13.

Answer 1

calculating remainders using CONGRUENCES is one of the methods when we cannot use the calculator to divide two numbers!

So first of all what is a congruence?

When we divide a number a with n, then using Euclidean algorithm there exist numbers b and k such that a=kn+b where b is the remainder() , so we can rewrite it using congruence modulo n ''

  

remember n has to be a positive integer!

For example, when we divide 24 by 7 , we write it as 24=7•3 + 3 using Euclidean algorithm and using congruences we write is as

  

• This congruence relation is true if and only if a and b leave the same non-negative remainder when divided by n.

Now some properties for n>1 and a,b,c,d are arbitrary integers

1. 
2. 
3.   
4. 

These are ones which we will use extensively!!

• Starting from the bottom seems like a wise idea --

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Now, after getting a hang of it , we proceed further!!

Now, we move up to the big questions which require just a little bit more attention!!

35346 is a big number on its own, so why don't we break it up into smaller numbers like in c and d parts? The only way that we know how to do this is write is prime factorisation!!

35346= 2 x 3 x 43 x 137

Now, its basically what we did in the last parts!

so let's get on with it--

So, we are sure that the answer posted is wrong!

Moving on

Again, I think you have posted the wrong answer!

​​​​​​​But if doubts still persist you can comment and I would be happy to help!!