Divide \( 137 \) by \( 5 \):
\(137 \div 5 = 27 \quad \text{(remainder } 2\text{)}\)

Divide \( 27 \) by \( 5 \):
\(27 \div 5 = 5 \quad \text{(remainder } 2\text{)}\)

Divide \( 5 \) by \( 5 \):
\(5 \div 5 = 1 \quad \text{(remainder } 0\text{)}\)

Divide \( 1 \) by \( 5 \):
\(1 \div 5 = 0 \quad \text{(remainder } 1\text{)}\)

Now, we collect the remainders in reverse order: \( 1, 0, 2, 2 \).

Thus, \( 137 \) in base \( 5 \) is represented as \( 1022_5 \).