1. **State the problem:** We need to find two positive integers whose sum is 60 and whose least common multiple (LCM) is 273. 2. **Recall the relationship between two numbers, their greatest common divisor (GCD), and LCM:** For two integers $a$ and $b$, $$a \times b = \text{GCD}(a,b) \times \text{LCM}(a,b).$$ 3. **Let the two integers be $x$ and $y$.** Given: $$x + y = 60$$ $$\text{LCM}(x,y) = 273$$ 4. **Express the product $x \times y$ in terms of GCD and LCM:** $$x \times y = \text{GCD}(x,y) \times 273$$ 5. **Check each pair from the options to see which satisfies both conditions:** - Option 1: 27 and 33 Sum: $27 + 33 = 60$ LCM: Find GCD first. $$\text{GCD}(27,33) = 3$$ $$x \times y = 27 \times 33 = 891$$ Check if $891 = 3 \times 273$: $$3 \times 273 = 819 \neq 891$$ So LCM is not 273. - Option 2: 15 and 45 Sum: $15 + 45 = 60$ GCD: $$\text{GCD}(15,45) = 15$$ Product: $$15 \times 45 = 675$$ Check if $675 = 15 \times 273$: $$15 \times 273 = 4095 \neq 675$$ So LCM is not 273. - Option 3: 30 and 30 Sum: $30 + 30 = 60$ GCD: $$\text{GCD}(30,30) = 30$$ Product: $$30 \times 30 = 900$$ Check if $900 = 30 \times 273$: $$30 \times 273 = 8190 \neq 900$$ So LCM is not 273. - Option 4: 39 and 21 Sum: $39 + 21 = 60$ GCD: $$\text{GCD}(39,21) = 3$$ Product: $$39 \times 21 = 819$$ Check if $819 = 3 \times 273$: $$3 \times 273 = 819$$ This matches perfectly. 6. **Verify the LCM for 39 and 21:** $$\text{LCM}(39,21) = \frac{39 \times 21}{\text{GCD}(39,21)} = \frac{819}{3} = 273$$ **Answer:** The two integers are 39 and 21.