Question

Find the least common multiple (LCM) of 8y^6+ 144y^5+ 640y^4 and 2y^4 + 40y^3 + 200y^2.

You can give your answer in its factored form.

Answer 1

LCM =
`8y^(4)(y+ 10)^(2)(y + 8)`

Explanation:

Making factors of
`8y^(6)+ 144y^(5)+ 640y^(4)`

Taking
`8y^(4)` common:

`\Rightarrow 8y^(4) (y^(2)+ 18y+ 80)`

Using factorization method:

`\Rightarrow 8y^(4) (y^(2)+ 10y + 8y + 80)\\\Rightarrow 8y^(4) (y (y+ 10) + 8(y + 10))\\\Rightarrow 8y^(4) (y+ 10)(y + 8))\\\Rightarrow \underline{2y^(2)} * 4y^(2) \underline{(y+ 10)}(y + 8)) ..... (1)`

Now, Making factors of
`2y^(4) + 40y^(3) + 200y^(2)`

Taking
`2y^(2)` common:

`\Rightarrow 2y^(2) (y^(2)+ 20y+ 100)`

Using factorization method:

`\Rightarrow 2y^(2) (y^(2)+ 10y+ 10y+ 100)\\\Rightarrow 2y^(2) (y (y+ 10) + 10(y + 10))\\\Rightarrow \underline {2y^(2) (y+ 10)}(y + 10) ............ (2)`

The underlined parts show the Highest Common Factor(HCF).

i.e. HCF is
`2y^(2) (y+ 10)`.

We know the relation between LCM, HCF of the two numbers 'p' , 'q' and the numbers themselves as:

`HCF * LCM = p * q`

Using equations (1) and (2):
`\Rightarrow 2y^(2) (y+ 10) * LCM = 2y^(2) * 4y^(2)(y+ 10)(y + 8) * 2y^(2) (y+ 10)(y + 10)\\\Rightarrow LCM = 2y^(2) * 4y^(2)(y+ 10)(y + 8) * (y + 10)\\\Rightarrow LCM = 8y^(4)(y+ 10)^(2)(y + 8)`

Hence, LCM =
`8y^(4)(y+ 10)^(2)(y + 8)`