What is the least common multiple of 30, 20, and 70

What is the least common multiple of 30, 20, and 70 ?

  1. 40
  2. 42 
  3. 120 
  4. 420 
  5. 42,000

Answer

The correct answer is D.

You can list the multiples of the largest of the 3 numbers (70) as a sequence and determine whether or not each succeeding term in the sequence is a multiple of both 20 and 30. 

  • 70 (multiple of neither)
  • 140 (multiple of 20 only)
  • 210 (multiple of 30 only)
  • 280 (multiple of 20 only)
  • 350 (multiple of neither)
  • 420 (multiple of both 20 and 30)

The first term in the sequence that is a multiple of both 20 and 30 is 420, which is the least common multiple of 20, 30, and 70.

Second Method

You can also find the least common multiple by expressing each of the three numbers as a product of primes (with exponents), listing all bases of exponential expressions and choosing for each base listed the highest-valued exponent.

30 = 21 × 31 × 51

20 = 22 × 51

70 = 21 × 51 × 71

The least common multiple is 22 × 31 × 51 × 71 = 420