Question

determine whether each of the following is true or false. explain why.

(a) \\(9 \equiv 17 \pmod{3}\\)

(b) \\(9 \equiv 17 \pmod{4}\\)

Explanation:

Define modular congruence

Two integers \(a\) and \(b\) are congruent modulo \(m\), written \(a \equiv b \pmod m\), if and only if their difference \(a - b\) is divisible by \(m\). This means \(m\) divides \(a - b\), denoted as \(m \mid (a - b)\).

Analyze statement (a)

We evaluate \(9 \equiv 17 \pmod 3\).
Calculate the difference:

$$ 9 - 17 = -8 $$

Check if \(3\) divides \(-8\):

$$ \frac{-8}{3} otin \mathbb{Z} $$

Since \(3\) does not divide \(-8\), the statement is false.

Analyze statement (b)

We evaluate \(9 \equiv 17 \pmod 4\).
Calculate the difference:

$$ 9 - 17 = -8 $$

Check if \(4\) divides \(-8\):

$$ \frac{-8}{4} = -2 \in \mathbb{Z} $$

Since \(4\) divides \(-8\), the statement is true.

Answer:

Question (a)

False. The difference \(9 - 17 = -8\) is not divisible by \(3\).

Question (b)

True. The difference \(9 - 17 = -8\) is divisible by \(4\).