Introduction to Number Theory: Remainders and Division
Number theory is a fundamental branch of mathematics that deals with properties and relationships of numbers, especially integers. A key concept in number theory is the remainder, which arises when a number is divided by another number. This article will delve into a specific problem: finding the remainder when a given number n is divided by 35, given that n leaves a remainder of 7 when divided by 28. We will use this problem to explore the division algorithm and congruences in number theory.
Problem and Approach
Consider a number n. According to the information provided, when n is divided by 28, it leaves a remainder of 7. Mathematically, this can be expressed as:
n 28k 7
where k is some integer. The goal is to find the remainder when n is divided by 35. To achieve this, we will use the division algorithm and properties of congruences.
Step-by-Step Solution
First, let's rewrite n in terms of its congruence modulo 35:
n 28k 7
Next, we need to find n modulo 35:
n mod 35 (28k 7) mod 35
We can simplify this expression by considering the congruence of 28 modulo 35:
28 ≡ -7 (mod 35)
Therefore:
28k ≡ -7k (mod 35)
Substituting back into the expression for n:
n mod 35 (-7k 7) mod 35
This can be further simplified to:
n mod 35 7 - 7k (mod 35)
To find the possible remainders, we need to analyze the term 7 - 7k for various values of k, where k is an integer.
Calculation for Various Values of k
Let's calculate the remainder for different values of k:
For k 0:n mod 35 7 - 7(0) 7 (mod 35)
For k 1:n mod 35 7 - 7(1) 0 (mod 35)
For k 2:n mod 35 7 - 7(2) 28 (mod 35)
For k 3:n mod 35 7 - 7(3) 21 (mod 35)
For k 4:n mod 35 7 - 7(4) 14 (mod 35)
For k 5:n mod 35 7 - 7(5) 7 (mod 35)
Notice that the remainder repeats every 5 values of k. Therefore, the possible remainders when n is divided by 35 are 7, 0, 28, 21, and 14, depending on the value of k.
Practical Example
Consider the example where the original number N is 35. If we divide 35 by 28, the remainder would be 7. Now, if we want to find the remainder when 35 is divided by 35:
35 ÷ 35 1 with a remainder of 0
This is a simple case that demonstrates the basic principle of division.
Final Conclusion
In conclusion, when a number n is divided by 28, leaving a remainder of 7, the remainder when n is divided by 35 can be one of the following: 7, 0, 28, 21, or 14, depending on the value of k.