The LCM of two different numbers is 18, and the GCF is 6. What are the numbers?

Question

Levi Adams · Accepted Answer

The numbers themselves are 6 and 18.

Each number could be 1, 2, 3, 6, 9, or 18 if the LCM of the two numbers is 18. This implies that both numbers must be factors of 18. Both numbers are divisible by six if their GCF is six. As a result, any number could be 6, 12, 18, 24,..., etc. When we overlap these two restrictions, we see that the only values common to both sets are 6 and 18. So our possible pairs are #(6, 6)#, #(6, 18)#, or #(18, 18)#. But wait—only one of these pairs has both an LCM of 18 and a GCF of 6. That pair is #(6, 18)#. So, this is our answer. In general, you may also be able to use the fact that the product of any two numbers will equal the product of #("their lcm") times ("their gcf")#. Let's show why with an example pair: 6 and 15. #6=2*3# #15=3 * 5# GCF: They both have just a 3 in common, so their GCF is 3. LCM: The first number that's a multiple of both will need to be a multiple of 2, 3, and 5. The least of such multiples is #2*3*5=30#. So we get #6 times 15# #=(2 * 3) * (3 * 5)# #=color(red)2 * color(blue)3 * color(red)3 * color(red)5# #=color(red)lcm(6,15) times color(blue)("gcf"(6, 15))# This method is always effective. However, for the numbers 6 and 18, which coincidentally happened to be the GCF and LCM already, it was not required nor particularly helpful.

Luna Angel · Answer

They must have the prime factors #2xx3=6# in common.

#18=2xx3xx3# and #6=2xx3# would do.

Gabriel Agustin · Answer

undefined To find the numbers given their least common multiple (LCM) and greatest common factor (GCF), you can use the relationship between the LCM, GCF, and the actual numbers.

If $ a $ and $ b $ are the two numbers, then:

$$ \text{LCM}(a, b) \times \text{GCF}(a, b) = a \times b $$

Given that the LCM is 18 and the GCF is 6, we can substitute these values into the equation:

$$ 18 \times 6 = a \times b $$

$$ 108 = a \times b $$

Now, we need to find two numbers that multiply to give 108 and whose GCF is 6. The pairs of numbers that meet this condition are 9 and 12.

So, the numbers are 9 and 12.