Question

How can you the least find denominator for 1/8 and 2/9?

Answer 1

`72`

Explanation:

First, find the prime factorization of each denominator:

`8 = 2xx2xx2 = 2^3`
`9 = 3xx3 = 3^2`

Next, find the product of the greatest powers of each prime that occurs:

`2^3xx3^2 = 8xx9 = 72`

In this case, the least common denominator is `72`.

`1/8 = (1xx9)/(8xx9) = 9/72`
`2/9 = (2xx8)/(9xx8) = 16/72`

---

In the above case, we get the same result by just multiplying the two denominators. For an example where that is not the case, consider `1/12` and `1/18`

`12 = 2xx2xx3 = 2^2xx3^1`
`18 = 2xx3xx3 = 2^1xx3^2`

The only primes which appear are `2` and `3`. The greatest power of `2` is `2^2`. The greatest power of `3` is `3^2`. Multiplying them, we get

`2^2 xx 3^2 = 4xx9 = 36`

So the least common denominator between `1/12` and `1/18` is `36`.

`1/12 = (1xx3)/(12xx3) = 3/36`
`1/18 = (1xx2)/(18xx2) = 2/36`