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Is the three-digit number n less than 550?

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Is the three-digit number n less than 550?

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Is the three-digit number n less than 550?

(1) The product of the digits in n is 30.
(2) The sum of the digits in n is 10.

OA C

Source: GMAT Prep

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BTGmoderatorDC wrote:
Is the (positive) three-digit number n less than 550?

(1) The product of the digits in n is 30.
(2) The sum of the digits in n is 10.

Source: GMAT Prep
Very nice problem!

\[100 \leqslant \,\,n = \left\langle {ABC} \right\rangle \,\, \leqslant 999\]
\[\left\langle {ABC} \right\rangle \,\,\mathop < \limits^? \,\,550\]
\[\left( 1 \right)\,\,\,A \cdot B \cdot C = 30\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {A,B,C} \right) = \left( {1,5,6} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\
\,{\text{Take}}\,\,\left( {A,B,C} \right) = \left( {5,6,1} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,\,A + B + C = 10\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {A,B,C} \right) = \left( {1,2,7} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\
\,{\text{Take}}\,\,\left( {A,B,C} \right) = \left( {5,5,0} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\end{gathered} \right.\]
\[\left( {1 + 2} \right)\,\,\,\left\langle {ABC} \right\rangle \,\,\, \geqslant \,\,\,550\,\,\,\, \Rightarrow \,\,\,\left\{ \begin{gathered}
\,A = 5\,\,,\,\,B \geqslant 5\,\,\,\mathop \Rightarrow \limits^{\left( 2 \right)} \,\,\,B = 5,\,\,C = 0\,\,\, \Rightarrow \,\,\,\left( 1 \right)\,\,{\text{contradicted}} \hfill \\
\,\,\,{\text{or}} \hfill \\
\,A \geqslant 6\,\,\,\mathop \Rightarrow \limits_{\left( 1 \right)}^{\left( * \right)} \,\,\,A = 6\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,B \cdot C = 5\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,A + B + C = 6 + 5 + 1\,\,\, \Rightarrow \,\,\,\left( 2 \right)\,\,{\text{contradicted}} \hfill \\
\end{gathered} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \]
\[\left( * \right)\,\,\,\left( 1 \right)\,\,\, \Rightarrow \,\,\,A,B,C\,\,{\text{are}}\,\,\left( {{\text{positive}}} \right)\,\,{\text{divisors}}\,\,{\text{of}}\,\,30\,\,\, \Rightarrow \,\,\,A \notin \left\{ {7,8,9} \right\}\]
\[\left( {**} \right)\,\,\,B \cdot C = 5\,\,\, \Rightarrow \,\,\,B,C\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{pair}}\,\,{\text{of}}\,\,\left( {{\text{positive}}} \right)\,\,{\text{divisors}}\,\,{\text{of}}\,\,5\,\,\, \Rightarrow \,\,\,\left\{ \begin{gathered}
\,\left( {B,C} \right) = \left( {1,5} \right) \hfill \\
\,\,{\text{or}} \hfill \\
\,\left( {B,C} \right) = \left( {5,1} \right) \hfill \\
\end{gathered} \right.\]

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

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BTGmoderatorDC wrote:
Is the three-digit number n less than 550?

(1) The product of the digits in n is 30.
(2) The sum of the digits in n is 10.

OA C

Source: GMAT Prep
Given: n is a three-digit number.

Question: Is n < 550?

Let's take each statement one by one.

(1) The product of the digits in n is 30.

30 = 1*2*3*5

Case 1: Say n = 615. The answer is no.
Case 2: Say n = 156. The answer is yes.

No unique answer. Insufficient.

(2) The sum of the digits in n is 10.

Case 1: Say n = 910. The answer is no.
Case 2: Say n = 190. The answer is yes.

No unique answer. Insufficient.

(1) and (2) together

Given the product of the digits = 30 = 1*2*3*5, we must have one of the digits equal to 5, and given that the sum of the digits equals to 10, the sum of other two digits equals to 10 - 5 = 5; thus, the other two digits must be 2 and 3.

With three digits 2, 3, and 5, we can have a maximum number = 532, which is less than 550. The answer is yes. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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