Let $a$ be an integer such that $a \equiv 5 \pmod{7}$. Find the value of $a + 1 \pmod{7}$. Express your answer as a residue between $0$ and the modulus.

Answer 1

To find the value of \( a + 1 \pmod{7} \) given \( a \equiv 5 \pmod{7} \), we start by noting the initial equivalence:

\[
a \equiv 5 \pmod{7}
\]

Adding 1 to both sides of this congruence:

\[
a + 1 \equiv 5 + 1 \pmod{7}
\]

This simplifies to:

\[
a + 1 \equiv 6 \pmod{7}
\]

Since 6 is already within the specified range of residues (0 to 6), we conclude:

\[
a + 1 \pmod{7} = 6
\]

Thus, the value of \( a + 1 \pmod{7} \) is:

\[
\boxed{6}
\]