Use the Euclidean algorithm to find the following. GCD \( (270,567) \) and \( \operatorname{LCM}(270,567) \) Note: "GCD" means greatest common divisor and "LCM" means least common multiple. \( \operatorname{GCD}(270,567)=\square \)

We start by using the Euclidean algorithm on \(270\) and \(567\): 1. Divide \(567\) by \(270\): \[ 567 = 270 \times 2 + 27. \] The remainder is \(27\). 2. Now, take the previous divisor \(270\) and divide it by the remainder \(27\): \[ 270 = 27 \times 10 + 0. \] The remainder is \(0\), so the algorithm stops here. Thus, \[ \operatorname{GCD}(270,567)=27. \] Next, we use the relationship between the greatest common divisor (GCD) and the least common multiple (LCM): \[ \operatorname{LCM}(a,b)=\frac{a \times b}{\operatorname{GCD}(a,b)}. \] Substitute \(a=270\), \(b=567\), and \(\operatorname{GCD}(270,567)=27\): \[ \operatorname{LCM}(270,567)=\frac{270 \times 567}{27}. \] Simplify by dividing \(270\) by \(27\): \[ \frac{270}{27}=10, \] so \[ \operatorname{LCM}(270,567)=10 \times 567 = 5670. \] Thus, the final answers are: \[ \operatorname{GCD}(270,567)=27 \quad \text{and} \quad \operatorname{LCM}(270,567)=5670. \]