Question

What is the LCM of 10, 15, 20 and 30?

Answer 1

`60`

Explanation:

First, write out the prime factorization of each number

• `10 = 2 xx 5`
• `15 = 3 xx 5`
• `20 = 2^2 xx 5`
• `30 = 2 xx 3 xx 5`

We can rewrite the above with more clarity as

• `10 = 2^1 xx 3^0 xx 5^color(blue)(1)`
• `15 = 2^0 xx 3^color(blue)(1) xx 5^color(blue)(1)`
• `20 = 2^color(blue)(2) xx 3^0 xx 5^color(blue)(1)`
• `30 = 2^1 xx 3^color(blue)(1) xx 5^color(blue)(1)`

For each prime factor, take the one with the highest exponent. 2 is raised to the power of 2 in 20. 3 and 5 have both a maximum exponent of 1. Refer to the `color(blue)("blue")` colored exponents above.

Therefore,

`"LCM" = 2^2 xx 3^1 xx 5^1`

`= 60`

This algorithm is guaranteed to generate the least common multiple.

Answer 2

`LCM = 60`

Explanation:

The first thing to notice is that we do not need to consider `10 and 15` at all because they are factors of `20 and 30` respectively.

We only need to find the LCM of `color(blue)(20 and 30)`

You should be very familiar with these two numbers and their multiples.

The quickest and easiest method is to consider the multiples of the bigger one (`30)`, until you find the first one which is a multiple of `20`.

The multiples of `30` are: `30, color(magenta)(60), 90, 120 ...`
`color(white)(wwwwwwwwww.wwwww)uarr`
`color(white)(wwwwwwwwww.wwww)20 xx3`

`60` is the multiple we need. It is divisible by `10,15,20 and 30`

If the given numbers had been bigger or with less obvious factors and multiples, then I would have used the method of prime factors, but this one can be found mentally.