1. **State the problem:** We need to find the lowest common multiple (LCM) of 30 and 42 by first drawing their prime factor trees. 2. **Prime factorization:** - For 30: Start dividing by the smallest prime 2, then 3, then 5. - For 42: Start dividing by 2, then 3, then 7. 3. **Prime factor tree for 30:** $$30 \rightarrow 2 \times 15$$ $$15 \rightarrow 3 \times 5$$ So, prime factors of 30 are $$2, 3, 5$$. 4. **Prime factor tree for 42:** $$42 \rightarrow 2 \times 21$$ $$21 \rightarrow 3 \times 7$$ So, prime factors of 42 are $$2, 3, 7$$. 5. **Formula for LCM using prime factors:** The LCM is found by taking the highest power of each prime factor appearing in either number. 6. **Apply the formula:** - Prime factors: 2, 3, 5, 7 - For 30: $$2^1, 3^1, 5^1$$ - For 42: $$2^1, 3^1, 7^1$$ 7. **Calculate LCM:** $$LCM = 2^1 \times 3^1 \times 5^1 \times 7^1 = 2 \times 3 \times 5 \times 7 = 210$$ 8. **Conclusion:** The lowest common multiple of 30 and 42 is **210**.