1. **State the problem:** Find the least common multiple (LCM) of the numbers 20, 66, 77, 88, and 99. 2. **Formula and rules:** The LCM of several numbers is the smallest positive integer divisible by all of them. 3. **Prime factorization:** - $20 = 2^2 \times 5$ - $66 = 2 \times 3 \times 11$ - $77 = 7 \times 11$ - $88 = 2^3 \times 11$ - $99 = 3^2 \times 11$ 4. **Find the highest powers of all primes appearing:** - For 2: highest power is $2^3$ (from 88) - For 3: highest power is $3^2$ (from 99) - For 5: highest power is $5^1$ (from 20) - For 7: highest power is $7^1$ (from 77) - For 11: highest power is $11^1$ (appears in 66, 77, 88, 99) 5. **Calculate the LCM:** $$\text{LCM} = 2^3 \times 3^2 \times 5 \times 7 \times 11$$ 6. **Evaluate step-by-step:** - $2^3 = 8$ - $3^2 = 9$ - Multiply: $8 \times 9 = 72$ - Multiply: $72 \times 5 = 360$ - Multiply: $360 \times 7 = 2520$ - Multiply: $2520 \times 11 = 27720$ **Final answer:** The LCM of 20, 66, 77, 88, and 99 is $27720$.