To determine if $2a + b \equiv 0 \pmod{3}$, let's analyze the expression.

What it means:

The congruence $2a + b \equiv 0 \pmod{3}$ implies that the remainder when $2a + b$ is divided by 3 is zero. In other words, $2a + b$ must be divisible by 3.

Steps to check:

1. Simplify $2a + b \mod 3$:

   • $2a + b \mod 3$ means the sum $2a + b$ is reduced modulo 3.
   • If $2a \mod 3$ and $b \mod 3$ are known, you can add the results and reduce modulo 3 to check if the sum equals 0.

2. Verify divisibility:

   • If $a$ and $b$ are integers, evaluate $2a \mod 3$ and $b \mod 3$ individually. Substitute into the equation to confirm the congruence.

Example:

• If $a = 1$ and $b = 1$, calculate: $2a + b = 2(1) + 1 = 3 \equiv 0 \pmod{3}.$ So the congruence holds for this example.

For a generalized answer:

• The congruence $2a + b \equiv 0 \pmod{3}$ holds true only if $2a + b$ is divisible by 3.

Let me know if you’d like me to evaluate this further or provide specific examples!

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5 Questions to Expand Understanding:

1. What values of $a$ and $b$ satisfy $2a + b \equiv 0 \pmod{3}$?
2. How does modular arithmetic affect solving linear equations?
3. Can we generalize this solution for other moduli (e.g., mod 4 or mod 5)?
4. What are some applications of modular arithmetic in real-world problems?
5. How can systems of congruences (e.g., simultaneous modular equations) be solved?

Tip:

Always simplify individual terms modulo $n$ before combining them in modular arithmetic—it makes calculations faster and reduces errors.