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.
What is the product of integers a, b, and c if...
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Hi LUANDATO,
You should notice that all of the 'bases' are PRIME numbers, so this question is all about 'prime factorization.' To start, we have to prime factor 270,000,000 down into its 'pieces'...
(270)(1,000)(1,000) =
270 = (3)(3)(3)(2)(5)
1,000 = (2)(2)(2)(5)(5)(5)
Thus 270,000,000 = (2^7)(3^3)(5^7)
The product of those three exponents is (7)(3)(7) = 147
Final Answer: B
GMAT assassins aren't born, they're made,
Rich
You should notice that all of the 'bases' are PRIME numbers, so this question is all about 'prime factorization.' To start, we have to prime factor 270,000,000 down into its 'pieces'...
(270)(1,000)(1,000) =
270 = (3)(3)(3)(2)(5)
1,000 = (2)(2)(2)(5)(5)(5)
Thus 270,000,000 = (2^7)(3^3)(5^7)
The product of those three exponents is (7)(3)(7) = 147
Final Answer: B
GMAT assassins aren't born, they're made,
Rich
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Hi LUANDATO,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.
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.
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Let's break 270,000,000 into its prime factors.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
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
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