What is the least common denominator of 6/16 and 1/15?

Jun 3, 2017

The least common denominator of $\frac{x}{16} \text{ and } \frac{x}{15}$ is $\frac{x}{240}$

Explanation:

To find the lowest common denominator, we need to find the lowest common multiple ($L C M$) of the two denominators.

To find the lowest common multiple of two numbers - in this case, $16$ and $15$, we need to find the prime factorisation of each number. We can do this either by entering the number in on a scientific calculator (most scientific calculators should have this function) and press the $\text{FACT}$ button, this will give you the prime factorisation of that number. You can also do it manually, which I am going to demonstrate here.

To find the prime factorisation of a number, we need to divide the number by the lowest number possible, then getting all numbers to prime numbers by dividing, again by the lowest number possible.

$16$
÷color(red)(2) = 8
÷color(red)(2) = 4
÷color(red)(2) = color(red)(2)

We do not divide until it is $1$, because the numbers are all already prime. we stop the process when all of the numbers are prime.

So we can now say that the numbers in red are the prime factorisations of $16$. Now we simplify them in a multiplication fashion.

$16 = 2 \times 2 \times 2 \times 2$

color(blue)(16 = 2^4

Now we can do the same thing to $15$

$15$
÷color(red)(3) = color(red)(5)

Because the numbers are now prime, the process is over.

color(blue)(15 = 3 xx 5

We can not simplify this number any further.

Now that we have the prime factorisations of each number, we can find the lowest common multiple of the numbers.

To find the Lowest common multiple, we will multiply all of the common numbers by the uncommon numbers.

For example:

$72 = \cancel{{2}^{3}} \times {3}^{2}$

$56 = \cancel{{2}^{3}} \times 7$

Because there are two sets of ${2}^{3}$, we cancel them out and use one of them in the equation.

$L C M = {2}^{3} \times {3}^{2} \times 7$

$L C M = 8 \times 9 \times 7$

$L C M = 504$

$16 = {2}^{4}$

$15 = 3 \times 5$

$L C M = {2}^{4} \times 3 \times 5$

$L C M = 16 \times 3 \times 5$

color(blue)(LCM = 240

$\therefore$ The lowest common denominator of $\frac{x}{16} \text{ and } \frac{x}{15}$ is $\frac{x}{240}$