To determine which residues satisfy the congruence \( 3x \equiv 0 \pmod{5} \), we start by recognizing that a number \( x \) satisfies this congruence if \( 3x \) is congruent to \( 0 \pmod{5} \). This equivalence can be written as:
\[
3x = 5k \text{ for some integer } k
\]
Here, we see that \( 3x \equiv 0 \pmod{5} \) implies that \( 3x \) must be a multiple of 5. Since \( 3 \) and \( 5 \) are coprime (their greatest common divisor is 1), \( x \) must be a multiple of \( \frac{5}{\text{gcd}(3, 5)} = 5 \). However, we are looking at residues modulo \( 5 \), so we need \( x \) within \( \{0, 1, 2, 3, 4\} \).
We test each residue:
1. For \( x = 0 \):
\[
3 \cdot 0 = 0 \equiv 0 \pmod{5}
\]
Thus, \( x = 0 \) satisfies the congruence.
2. For \( x = 1 \):
\[
3 \cdot 1 = 3 \not\equiv 0 \pmod{5}
\]
Thus, \( x = 1 \) does not satisfy the congruence.
3. For \( x = 2 \):
\[
3 \cdot 2 = 6 \equiv 1 \pmod{5}
\]
Thus, \( x = 2 \) does not satisfy the congruence.
4. For \( x = 3 \):
\[
3 \cdot 3 = 9 \equiv 4 \pmod{5}
\]
Thus, \( x = 3 \) does not satisfy the congruence.
5. For \( x = 4 \):
\[
3 \cdot 4 = 12 \equiv 2 \pmod{5}
\]
Thus, \( x = 4 \) does not satisfy the congruence.
The only residue that satisfies the congruence \( 3x \equiv 0 \pmod{5} \) is:
\[
\boxed{0}
\]
Which of the residues $0,$ $1,$ $2,$ $3,$ $4$ satisfy the congruence $3x \equiv 0 \pmod{5}?$
Give your answer as a list, separated by commas, in order from least to greatest.
1 answer