Number Theory Exploration: Possible Remainders with Division by 357

In number theory, understanding how a number behaves when divided by a certain divisor can reveal fascinating patterns and relationships. One such problem involves a number, when divided by 17, leaving a remainder of 5. We will explore how this number behaves when divided by 357 and identify the possible remainders.

Understanding the Problem

Let the number be ( n ). According to the problem, we have:

n ≡ 5 mod 17

From this, we can express ( n ) in the form:

n 17k 5

for some integer ( k ). This expression simply states that when ( n ) is divided by 17, the remainder is 5.

Dividing by 357: A Closer Look

Next, we need to find the remainder when ( n ) is divided by 357. Let's analyze the term ( n 17k 5 ) modulo 357. This can be broken down into:

n mod 357 (17k 5) mod 357

Simplifying further, we can express this as:

n mod 357 (17k mod 357) (5 mod 357)

Since ( 5

5 mod 357 5

Therefore, to find the possible remainders, we need to examine how the term ( 17k mod 357 ) behaves.

Examining Values of ( k )

Let's consider the behavior of ( 17k ) modulo 357 by calculating the remainders for several values of ( k ):

For ( k 0 ):
n 17(0) 5 5
5 mod 357 5 For ( k 1 ):
n 17(1) 5 22
22 mod 357 22 For ( k 2 ):
n 17(2) 5 39
39 mod 357 39

This pattern continues, with each successive value of ( k ) increasing the remainder by 17:

For ( k 3 ):
n 17(3) 5 56
56 mod 357 56 For ( k 4 ):
n 17(4) 5 73
73 mod 357 73 For ( k 5 ):
n 17(5) 5 90
90 mod 357 90

Continuing this process, we observe the sequence:

Value of k Remainder when n is divided by 357 0 5 1 22 2 39 3 56 4 73 5 90 6 107 7 124 8 141 9 158 10 175 11 192 12 209 13 226 14 243 15 260 16 277 17 294 18 311 19 328 20 345 21 362 ≡ 5 (mod 357)

Conclusion

The possible remainders when dividing the number ( n ) by 357 are:

5 22 39 56 73 90 107 124 141 158 175 192 209 226 243 260 277 294 311 328 345

The remainders range from 5 to 345, with increments of 17.