Finding the least common denominator

Explanation

Both ways are correct.  To explain it, let's find the least common denominator of \(\frac38\) and \(\frac{1}{10}\).  8 and 10 must divide the LCD exactly without remainders.  Thus the least common denominator of \(\frac38\) and \(\frac{1}{10}\) is simply the least common multiple of 8 and 10.

Method 1: Write the multiples of the largest denominator in increasing order, until one is found that is divisible by the other denominators.

We can find the LCM of 8 and 10 by listing multiples of the larger number, 10, until we find one that is divisible by the smaller number, 8.

Multiples of 10: 10, 20, 30, 40, 50, 60

40 is the first multiple of 10 that is divisible by 8 (no remainder)

Since the LCM of 8 and 10 is 40, it follows that the LCD of \(\frac38\) and \(\frac{1}{10}\) is 40.

Method 2: Prime factor each denominator. The LCM is a product of prime factors, where each factor is used the greatest number of times it appears in any one factorization.

We can also find the LCM of 8 and 10 using prime factorization.  We begin by prime factoring 8 and 10.

8 = 2 • 2 • 2

10 = 2 • 5

The LCM of 8 and 10 is a product of prime factors, where each factor is used the greatest number of times it appears in any one factorization.

• We will use the factor 2 three times, because 2 appears three times in the factorization of 8. 
• We will use the factor 5 once, because it appears one time in the factorization of 10

LCM(8, 10) = 2 • 2 • 2 • 5 = 40