How many factors of $2400$ are not factors of $3600$?

Question

I solved this question by writing all the factors and then just selecting the factors as per the question requirement.

But I want to know is there any other way to solve this? Please help !!!

Thanks in advance !!!

Answer 1

If you do a prime number decomposition you find

$$2400 = 2^5\cdot 3 \cdot 5^2$$ $$3600 = 2^4\cdot 3^2 \cdot 5^2$$

So the only time you can have a factor of $2400$ and not of $3600$ is when $2^5=32$ divides it.

Thus, the factors are $$ 32=2^5$$ $$ 96=2^5 \cdot 3$$ $$ 480=2^5 \cdot 3 \cdot 5$$ $$ 160=2^5 \cdot 5$$ $$ 800=2^5 \cdot 5^2$$ $$ 2400=2^5 \cdot 3 \cdot 5^2$$