Lcm and hcf question grade 7

Question

I am a a student and I am having difficulty with answering this question. I keep getting the answer wrong. Please may I have a step by step solution to this question so that I won't have difficulties with answering these type of questions in the future.

n is a number. 100 is the LCM of 20 and n. Work out two different possible values for n.

n = ______ n = ______

I did this: Hcf of 100:1,2,4,5,10,20,25,50,100

I don't know what to do next?

Thank you and help would be appreciated

Answer 1

First of all, get the divisors of 20:

20 is divisible by 1,2,4,5,10,20

Then let's look at the divisors of 100:

100 is divisible by 1,2,4,5,10,20,25,50, 100.

The only possible solutions are 25, 50 and 100 because n can not be a divisor of 20.

Answer 2

$20=2^2×5$

Suppose $n$ is of the form
$2^a×5^b$ where $a \ge 2; b\ge1$

Then $2^a×5^b=100$ $\implies a=2,b=2 \implies n=100$

Suppose $n$ is of the form
$5^b$ where $b\ge1$

Then $2^2×5^b=100$ $\implies b=2 \implies n=25$