Understanding Remainders and Modular Arithmetic

This problem involves finding the remainder when a multiple of a number is divided by a specific divisor, given the remainder of the original number. This concept is best understood using modular arithmetic.

The problem states that when an integer \(m\) is divided by 7, the remainder is 5. In the language of modular arithmetic, this can be written as:

\(m \equiv 5 \pmod{7}\)

This means that \(m\) can be expressed in the form \(m = 7k + 5\) for some integer \(k\).

Finding the Remainder of 3m

We need to find the remainder when \(3m\) is divided by 7. Let's use the property of congruences that allows multiplication. If \(a \equiv b \pmod{n}\), then \(ca \equiv cb \pmod{n}\) for any integer \(c\).

Since we know \(m \equiv 5 \pmod{7}\), we can multiply both sides of the congruence by 3:

\(3m \equiv 3 \times 5 \pmod{7}\)

Now, calculate the product on the right side:

\(3m \equiv 15 \pmod{7}\)

To find the remainder when \(3m\) is divided by 7, we need to find the remainder when 15 is divided by 7.

We can perform the division:

\(15 = 2 \times 7 + 1\)

The remainder when 15 is divided by 7 is 1.

Therefore, in terms of modular arithmetic:

\(15 \equiv 1 \pmod{7}\)

Substituting this back into our congruence for \(3m\):

\(3m \equiv 15 \equiv 1 \pmod{7}\)

This tells us that when \(3m\) is divided by 7, the remainder is 1.

Step-by-Step Calculation

Here is a summary of the steps:

1. Understand the given information: \(m\) divided by 7 gives a remainder of 5.
2. Write this using modular notation: \(m \equiv 5 \pmod{7}\).
3. Determine what needs to be found: the remainder when \(3m\) is divided by 7.
4. Multiply the congruence by 3: \(3m \equiv 3 \times 5 \pmod{7}\).
5. Calculate the product: \(3m \equiv 15 \pmod{7}\).
6. Find the remainder of the new number (15) when divided by the original divisor (7): \(15 = 2 \times 7 + 1\).
7. The remainder is 1.
8. Conclude that \(3m \equiv 1 \pmod{7}\).

Using the Algebraic Form

Alternatively, we can use the algebraic form \(m = 7k + 5\).

Multiply \(m\) by 3:

\(3m = 3(7k + 5)\)

Distribute the 3:

\(3m = 21k + 15\)

We want to find the remainder when \(3m\) is divided by 7. We can rewrite the expression for \(3m\) to explicitly show division by 7:

\(3m = 21k + 15\)

Since \(21k\) is a multiple of 7 (\(21k = 7 \times 3k\)), the remainder when \(21k\) is divided by 7 is 0.

So, the remainder of \(3m\) when divided by 7 is the same as the remainder of 15 when divided by 7.

\(15 = 2 \times 7 + 1\)

The remainder is 1. Thus, \(3m = 7 \times (3k + 2) + 1\). This confirms that when \(3m\) is divided by 7, the remainder is 1.

Revision Table: Key Concepts for Remainder Problems

Additional Information: Exploring Modular Arithmetic

Modular arithmetic is a fundamental concept in number theory and has applications in various fields like cryptography, computer science, and clock arithmetic.

The set of possible remainders when dividing by 7 is \(\{0, 1, 2, 3, 4, 5, 6\}\). All integers fall into one of these 7 congruence classes modulo 7.

• Any integer \(n\) such that \(n \equiv 0 \pmod{7}\) is a multiple of 7.
• Any integer \(n\) such that \(n \equiv 1 \pmod{7}\) is one more than a multiple of 7.
• And so on, up to \(n \equiv 6 \pmod{7}\), which is one less than a multiple of 7 (or equivalently, \(-1 \pmod{7}\)).

In our problem, \(m \equiv 5 \pmod{7}\). This means \(m\) could be 5, 12, 19, 26, etc.

Let's check for \(m=5\): \(3m = 3 \times 5 = 15\). \(15\) divided by 7 is 2 with a remainder of 1.

Let's check for \(m=12\): \(3m = 3 \times 12 = 36\). \(36\) divided by 7 is 5 with a remainder of 1 (\(36 = 5 \times 7 + 1\)).

Let's check for \(m=19\): \(3m = 3 \times 19 = 57\). \(57\) divided by 7 is 8 with a remainder of 1 (\(57 = 8 \times 7 + 1\)).

As you can see, regardless of the specific value of \(m\) (as long as it satisfies \(m \equiv 5 \pmod{7}\)), the remainder when \(3m\) is divided by 7 is always 1. Modular arithmetic provides a concise way to arrive at this general result without testing specific values of \(m\).