What is the product of integers a, b, and c if...

What is the product of integers a, b, and c if
$$2^a\ \cdot\ 3^b\ \cdot\ 5^c\ =\ 270,000,000$$

A. 141
B. 147
C. 162
D. 235
E. 270

The OA is B.

Please, can any expert assist me with this PS question? I need help to solve it. Thanks.

Hi LUANDATO,
Lets take a look at your question.

$$2^a\ \cdot\ 3^b\ \cdot\ 5^c\ =\ 270,000,000$$
$$2^a.3^b.5^c=27\times10,000,000$$ $$2^a.3^b.5^c=\left(3\times3\times3\right)\times\left(2\times5\right)\times\left(2\times5\right)\times\left(2\times5\right)\times\left(2\times5\right)\times\left(2\times5\right)\times\left(2\times5\right)\times\left(2\times5\right)$$
$$2^a.3^b.5^c=3^3\times2^7\times5^7$$
$$a=3,\ b=7,\ c=7$$
$$abc=3\times7\times7$$
$$abc=147$$

Therefore, option B is correct.

I am available if you'd like any follow up.

LUANDATO wrote:What is the product of integers a, b, and c if
$$2^a\ \cdot\ 3^b\ \cdot\ 5^c\ =\ 270,000,000$$

A. 141
B. 147
C. 162
D. 235
E. 270

Let’s break 270,000,000 into its prime factors.

270,000,000 = 27 x 10,000,000 = 3^3 x 10^7 = 2^7 x 3^3 x 5^7, so a x b x c = 7 x 3 x 7 = 147.

Answer: B