Understanding the Problem: Remainder with Division

The question asks us to find the remainder when a number \(2n\) is divided by 4, given that when another number \(n\) is divided by 4, the remainder is 3.

This type of problem deals with the concept of modular arithmetic, specifically focusing on remainders after division.

Expressing the Given Information Mathematically

When a number \(n\) is divided by 4, the remainder is 3. According to the division algorithm, this can be written as:

\(n = 4 \times q + 3\)

Here, \(q\) represents the quotient, which is an integer. This equation means that \(n\) is 3 more than a multiple of 4.

Finding the Expression for 2n

We need to find the remainder when \(2n\) is divided by 4. First, let's express \(2n\) using the equation we have for \(n\):

\(2n = 2 \times (4q + 3)\)

Now, distribute the 2:

\(2n = 2 \times 4q + 2 \times 3\)

\(2n = 8q + 6\)

Dividing 2n by 4 to Find the Remainder

Now we need to find the remainder when \(8q + 6\) is divided by 4. We can rewrite \(8q + 6\) to see the multiple of 4 more clearly.

Notice that \(8q\) is already a multiple of 4 because \(8q = 4 \times (2q)\). So, \(8q\) divided by 4 has a remainder of 0.

We can rewrite \(6\) as a multiple of 4 plus a remainder:

\(6 = 4 \times 1 + 2\)

Now substitute this back into the expression for \(2n\):

\(2n = 8q + (4 \times 1 + 2)\)

Group the terms that are multiples of 4:

\(2n = (8q + 4 \times 1) + 2\)

\(2n = 4 \times (2q + 1) + 2\)

This equation is in the form \(2n = 4 \times (\text{integer quotient}) + \text{remainder}\).

Here, the integer quotient is \(2q + 1\), and the remainder is 2.

Therefore, when \(2n\) is divided by 4, the remainder is 2.

Step-by-Step Calculation

1. Understand the given: \(n\) divided by 4 has remainder 3.
2. Write this using the division algorithm: \(n = 4q + 3\) for some integer \(q\).
3. Calculate \(2n\) by multiplying the expression for \(n\) by 2: \(2n = 2(4q + 3) = 8q + 6\).
4. Divide \(2n = 8q + 6\) by 4 to find the remainder.
5. Rewrite \(8q + 6\) as \(4 \times (2q) + 4 \times 1 + 2\).
6. Factor out 4: \(4 \times (2q + 1) + 2\).
7. The expression is now in the form \(4 \times (\text{quotient}) + \text{remainder}\).
8. The remainder is 2.

Checking with an Example

Let's pick a value for \(n\) that satisfies the condition. If \(n\) is divided by 4, the remainder is 3. Possible values for \(n\) are 3, 7, 11, 15, etc.

Let's take \(n = 7\). When 7 is divided by 4, \(7 = 4 \times 1 + 3\). The remainder is 3. This matches the condition.

Now, let's find \(2n\): \(2n = 2 \times 7 = 14\).

Now, divide \(14\) by 4:

\(14 = 4 \times 3 + 2\)

The remainder is 2.

Let's take another value, \(n = 11\). When 11 is divided by 4, \(11 = 4 \times 2 + 3\). The remainder is 3.

Now, let's find \(2n\): \(2n = 2 \times 11 = 22\).

Now, divide \(22\) by 4:

\(22 = 4 \times 5 + 2\)

The remainder is 2.

The examples confirm our general calculation that the remainder is 2.

Conclusion

Based on both the algebraic derivation and numerical examples, the remainder when \(2n\) is divided by 4, given that the remainder when \(n\) is divided by 4 is 3, is 2.

Revision Table

Additional Information: The Division Algorithm and Remainders

The division algorithm is a fundamental concept in number theory. It states that for any integer \(a\) (the dividend) and any positive integer \(b\) (the divisor), there exist unique integers \(q\) (the quotient) and \(r\) (the remainder) such that:

\(a = bq + r\)

where \(0 \le r < b\).

In our problem, \(a=n\), \(b=4\), and we were given that the remainder \(r=3\). So, \(n = 4q + 3\).

Remainders are important because they tell us about the properties of numbers related to division. For example, a number's remainder when divided by 2 tells us if it's even (remainder 0) or odd (remainder 1).

Properties of remainders can often simplify calculations. For example, if we know the remainders of two numbers when divided by \(m\), we can often find the remainder of their sum, difference, or product when divided by \(m\) without calculating the actual sum, difference, or product first. This is the basis of modular arithmetic.

In this question, we effectively used the properties of remainders: If \(n \equiv 3 \pmod{4}\), then \(2n \equiv 2 \times 3 \pmod{4}\). \(2 \times 3 = 6\). So, \(2n \equiv 6 \pmod{4}\). Now we find the remainder of 6 when divided by 4: \(6 = 4 \times 1 + 2\). So, \(6 \equiv 2 \pmod{4}\). Therefore, \(2n \equiv 2 \pmod{4}\), which means the remainder when \(2n\) is divided by 4 is 2.

This modular arithmetic approach confirms the result obtained through the algebraic substitution method.