Question

What is the least common multiple of `6,9, and 10`?

Answer 1

1

Explanation:

The only number 6, 10, and 9 can be divided by without a decimal is 1. This can be displayed by trying all the factors of one number, and trying them with the others. Let's try using 6.

Here are all of 6's factors:
`6 -: 6 = 1`
`6 -: 3 = 2`
`6 -:2 = 3`
`6 -:1 = 6`

1, 2, 3 or 6.

Now try to divide another number by each of these. Let's try 10.

`10 -: 1 = 10 sqrt`
`10 -: 2 = 5 sqrt`
`10 -: 3 = 3.333... xx`
`10 -: 6 = 1.666... xx`

Both 1 and 2 worked. Now, we try this with 9.

`9 -: 1 = 9 sqrt`
`9 -: 2 = 4.5 xx`

The only number left is 1, so this is your LCM.

Answer 2

`90`

Explanation:

one approach is to list the multiples and pick out the common ones

multiples of

`6:{6,12,18,24,30,36,42,48,54,60,66,72,78,84,color(red)(90),96,..}`

multiples of

`9:{9,18,27,36,45,54,63,72,81,color(red)(90),99,...}`

multiples of

`10:{10,20,30,40,50,60,70,80,color(red)(90),100,..}`

from the lists one can see that the

least common multiple is `90`

Answer 3

`LCM = 90`

Explanation:

Note that of the numbers given, `9 and 10` are consecutive numbers.

The LCM of two consecutive numbers is always their product.

So without other working we can immediately consider

`9xx10=90" "` as the LCM.

But will this work for `6` as well?

`6 =2xx3`

`3` is a factor of `9` and `2` is a factor of `10`, so `90` is also a multiple of `6`

Answer 4

The LCM of `6`, `9`, and `10` is `color(purple)90`.

Explanation:

Another method for determining the least common multiple (LCM) is using prime factorization. List the prime factors for each number. Then multiply each prime factor the greatest number of times it appears in any one factorization.

Find the LCM of `6`, `9`, and `10` by listing the prime factors of each number.

`6:` `color(red)2xx3`

`9:` `color(blue)3xxcolor(blue)3`

`10:` `2xxcolor(green)5`

Multiply each prime factor the greatest number of times it appears in any one factorization.

LCM: `color(red)2xxcolor(blue)3xxcolor(blue)3xxcolor(green)5=90`