# Problem of the Day: 1/11/13

Two cars begin at the same intersection. The red car drives north at 60 miles per hour on a straight road. The blue car drives east at 80 miles per hour on another straight road. After how many hours will the cars be exactly 200 miles apart?

Solution to yesterday’s problem:

Here’s an straightforward way to find a lowest common multiple of any two numbers:

1. Find the prime factorization of each number using a factor tree or method of your choice.
2. Strike out prime factors that are also in the other number’s factorization. For instance, if there are 3 fives in one factorization, and 2 fives in the other, strike out 2 fives total (it doesn’t matter which ones).
3. Multiply all the prime factors that are not canceled to get the LCM.

Applying this to our problem, we first need to find lcm(8,12). This problem is a simple example of the technique, but this method will also work for much more complex problems.

$8=2\times2\times2$

$12=2\times2\times3$

We can strike through the twos in 12’s factorization, leaving $2\times2\times2\times3$. So the LCM is 24. Next, we’ll find lcm(9,12).

$9=3\times3$

$12=2\times2\times3$

We can strike through the three in 12’s factorization, leaving $3\times3\times2\times2$. So the LCM is 36. Finally, we must find lcm(24,36). We already have the factorizations from above.

$24=2\times2\times2\times3$

$36=3\times3\times2\times2$

We should strike through the twos in 36’s factorization and the three in 24’s factorization, leaving $2\times2\times2\times3\times3$. The final LCM is 72.