Question

This handy application of LCMs is used by astronomers. All the planets in our solar system revolve around the sun. The planets occasionally line up together in their journeys, as shown in the illustration. The chart shows the time it takes each planet to make one trip around the sun. Now, imagine that the planets Earth, Mars, Jupiter, Saturn, Uranus, and Neptune aligned last night. How many years will pass before this happens again? (Hint—Find the LCM of the planets’ revolution times.) Solar System Planet Revolution Time Earth 1 year Mars 2 years Jupiter 12 years Saturn 30 years Uranus 84 years Neptune 165 years

Answer 1

To find number of years we take LCM of planets revolution times.

Planet's revolution times are given in the chart

Solar System Planet Revolution Time

Earth 1 year

Mars 2 years

Jupiter 12 years

Saturn 30 years

Uranus 84 years

Neptune 165 years

We take LCM of 1,2,12,30,84 and 165

Prime factorization of 2 = 2

Prime factorization of 12 = 2^2 * 3

Prime factorization of 30 = 2 * 3* 5

Prime factorization of 84 = 2^2 * 3 * 7

Prime factorization of 165 = 3*5*11

We take set of prime factors that has highest exponent,

So LCM = 2^2 * 3 * 5 * 7 * 11 = 4620

Therefore it will take 4620 years.