In number theory, a fundamental concept is the remainder when a number is divided by another. This article delves into the problem of finding the remainder when an integer (n) that leaves a remainder of 2 when divided by 7 is multiplied by 3, and then divided by 7. We'll explore this through a detailed mathematical proof and provide an intuitive understanding of the problem.

Introduction to Divisibility and Remainders

When a number (p) is divided by another number (k), we can express this division as (p kx r), where (x) is the quotient and (r) is the remainder. This is the foundational concept in divisibility and is often referred to as the division algorithm. For the specific case at hand, we know that (n div 7) leaves a remainder of 2. Mathematically, this can be written as:

(n 7k 2)

Multiplying n by 3n and Finding the Remainder when Divided by 7

To find the remainder when (n times 3n) is divided by 7, we start by expressing (n) in terms of its divisibility. Substituting (n 7k 2) into (n times 3n), we get:

(n times 3n (7k 2) times 3(7k 2))

First, let's simplify the expression:

(3n^2 3(7k 2)^2 3(49k^2 28k 4) 147k^2 84k 12)

Next, we need to find the remainder when (147k^2 84k 12) is divided by 7. We can reduce each term modulo 7:

147k^2 mod 7 0, since 147 is a multiple of 7. 84k mod 7 0, since 84 is a multiple of 7. 12 mod 7 5.

Therefore, combining these results, we get:

(147k^2 84k 12 equiv 0 0 5 mod 7 equiv 5 mod 7)

Hence, the remainder when (n times 3n) is divided by 7 is 5.

Additionally, let's explore another approach to the problem by considering the smallest possible values for (n). If (n / 7 k , r), with (n 7k 2), then when we multiply by 3, we get:

(3n 3(7k 2) 21k 6)

This shows that 3n also leaves a remainder of 6 when divided by 7. This can be verified by substituting small values of (k):

For (k 1), (n 9) (remainder 2 when divided by 7), and (3n 27) (remainder 6 when divided by 7). For (k 2), (n 16) (remainder 2 when divided by 7), and (3n 48) (remainder 6 when divided by 7).

In both cases, the remainder is 6, confirming the result.

Conclusion

In conclusion, when an integer (n) leaves a remainder of 2 when divided by 7, the remainder when (n times 3n) is divided by 7 is 5. Similarly, the remainder when (3n) is divided by 7 is 6. These results are derived from the properties of divisibility and remainders in number theory.