100 = 2² × 5², 150 = 2 × 3 × 5², 200 = 2³ × 5².

Common factors: 2¹ × 5² = 2 × 25 = 50.

This question asks us to find the least common multiple (LCM) of two numbers using prime factorization.

Break 84 into its prime factors by dividing by smallest primes.

Break 140 into its prime factors by dividing by smallest primes.

LCM is found by taking the highest power of each prime factor that appears in either number.

The LCM of 84 and 140 is 420, which is option (A).

A's rate = 1/12, B's rate = 1/18.

Combined rate = 1/12 + 1/18 = 3/36 + 2/36 = 5/36.

Time = 36/5 = 7.2 days

Combined rate = 1/10 + 1/15 + 1/30 = 3/30 + 2/30 + 1/30 = 6/30 = 1/5.

Time = 5 days

A completes 1/3 work in 5 days, so rate = 1/15 per day.

Remaining work = 2/3.

Days needed = (2/3)/(1/15) = (2/3) × 15 = 10 days

Combined rate = 1/8, A's rate = 1/12. B's rate = 1/8 - 1/12 = 3/24 - 2/24 = 1/24. B takes 24 days

9+8+7+6+5+4+3 = 42.

Then 4+2 = 6.

Wait, let me recalculate: Sum = 42, which reduces to 4+2=6.

The answer should be D.

Actually 9+8+7+6+5+4+3=42, 4+2=6.

Let numbers be 5x and 7x.

Then 5x + 7x = 120, so 12x = 120, x = 10.

Larger number = 7 × 10 = 70

Odd numbers between 10 and 30: 11, 13, 15, 17, 19, 21, 23, 25, 27, 29.

Count = 10

Let number be n.

Then n(n+1) = 342.

Solving: n² + n - 342 = 0.

Using quadratic formula or testing: 17 × 18 = 306 (no), 18 × 19 = 342 (yes).

So n = 18.

Wait, checking: 17 × 18 = 306, 18 × 19 = 342.

Answer is B.