1. **State the problem:** We need to find the largest square paving slab side length that can cover a rectangular patio of dimensions 390 cm by 180 cm without cutting or gaps. 2. **Formula and concept:** The largest square side length that fits perfectly into both dimensions is the greatest common divisor (GCD) of 390 and 180. 3. **Calculate the GCD:** - Prime factorization of 390: $390 = 2 \times 3 \times 5 \times 13$ - Prime factorization of 180: $180 = 2^2 \times 3^2 \times 5$ - Common prime factors: $2$, $3$, and $5$ - Take the lowest powers: $2^1$, $3^1$, $5^1$ - Multiply: $2 \times 3 \times 5 = 30$ 4. **Interpretation:** The largest square slab side length is $30$ cm. 5. **Verification:** - $390 \div 30 = 13$ slabs along width - $180 \div 30 = 6$ slabs along height - No cutting or gaps needed. **Final answer:** The largest side length of the paving slabs is **30 cm**.