How do you find the Least common multiple of #30ab^3, 20ab^3#?

Question

Noah Archer · Accepted Answer

The LCM is #60ab^3# #ab^3# is in both of them so we only need to look at the numbers to determine LCM. #color(blue)("Condition 1")# The last digit in both 30 and 20 is 0. So the multiple must also end in 0. #color(blue)("Condition 2")# The first digits are 3 and 2. The 2 means that they both have to be factors of an even number. The closest even number they both divide exactly into is 6. #color(blue)("Combining condition 1 and 2")# 6 put with 0 gives 60 Thus the LCM is #60ab^3#

Aurora Arias · Answer

The LCM is #60ab^3#.

The product of the largest amounts of each (prime) factor that appears in either number is the least common multiple (LCM) of two numbers; in other words, it is the smallest value that we can be certain will have both numbers as factors. First, factor the two numbers. #30ab^3# has the factors #[(2,3,5,a,b^3)]#. #20ab^3# has the factors #[(2^2, , 5, a, b^3)]#. Step 2: Circle the larger factor after comparing the powers of each element that appears in the two numbers. The factor #2# appears once in #30ab^3#, and it appears twice in #20ab^3#. Circle the #2^2#. #[(2,3,5,a,b^3),( (2^2), , 5, a, b^3)]# The factor #3# appears once in #30ab^3#, and not at all in #20ab^3#. Circle the #3#. #[(2,(3),5,a,b^3),( (2^2), , 5, a, b^3)]# The remaining three factors #(5, a, b^3)# appear the same number of times in both numbers. Circle either appearance of these factors. #[(2,(3),5,a,b^3),( (2^2), , (5), (a), (b^3))]# Step 3: Add up all of the values that are circled. The circled values are #(2^2)(3)(5)(a)(b^3)#. The product of these values is #60ab^3# Our least common multiple is this one.

Benjamin Altschul · Answer

undefined To find the least common multiple (LCM) of 30ab^3 and 20ab^3, you need to identify the highest powers of each prime factor that appear in either expression and multiply them together.

The prime factorization of 30ab^3 is $2 \times 3 \times 5 \times a \times b^3$.

The prime factorization of 20ab^3 is $2^2 \times 5 \times a \times b^3$.

To find the LCM, take the highest power of each prime factor that appears in either expression:

- For 2: The highest power is $2^2$.
- For 3: It only appears in 30ab^3, so it's $3^1$.
- For 5: The highest power is $5^1$.
- For a: It appears in both, so it's $a^1$.
- For $b^3$: The highest power is $b^3$.

Now, multiply these highest powers together:

$2^2 \times 3^1 \times 5^1 \times a^1 \times b^3 = 4 \times 3 \times 5 \times a \times b^3 = 60ab^3$.

Therefore, the least common multiple of 30ab^3 and 20ab^3 is 60ab^3.