The expression "6 modulo 7" is often interpreted with the remainder being non-negative in most mathematical contexts, which yields a result of $6$ because:

$6 \div 7 = 0 \quad \text{with a remainder of} \quad 6.$

Thus, in the standard non-negative modulo operation, we get:

$6 \mod 7 = 6.$

However, if we consider the modulo operation in a way that can yield a negative result, we can say:

$6 \equiv -1 \pmod{7}.$

This is because $6$ and $-1$ differ by a multiple of $7$ (specifically, $6 - (-1) = 7$), which means they are congruent modulo 7. In modular arithmetic, two numbers $a$ and $b$ are congruent modulo $n$ if $a - b$ is a multiple of $n$. So:

$6 \equiv -1 \pmod{7}.$

In summary:

• $6 \mod 7 = 6$ if we restrict to non-negative remainders.
• $6 \equiv -1 \pmod{7}$ when considering congruences, meaning $6$ behaves as $-1$ under modulo $7$.

Would you like further examples or a breakdown of modular arithmetic rules?

Here are 5 questions to expand your understanding:

1. What is the result of $-2 \mod 7$?
2. How do positive and negative results in modular arithmetic affect calculations?
3. What are common uses of modular arithmetic in real-life applications?
4. Can you determine if $15 \equiv -2 \pmod{17}$ holds true?
5. What is the relationship between modular congruence and division?

Tip: In modular arithmetic, think of congruent numbers as belonging to the same "class" or group within a repeating cycle of integers.