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# Is the three-digit number n less than 550?

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Manager
Joined: 16 Feb 2012
Posts: 175
Concentration: Finance, Economics
Is the three-digit number n less than 550?  [#permalink]

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15 Jun 2012, 05:34
3
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Difficulty:

65% (hard)

Question Stats:

61% (01:01) correct 39% (00:58) wrong based on 358 sessions

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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.

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Re: Is the three-digit number n less than 550?  [#permalink]

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15 Jun 2012, 05:39
6
Is the three-digit number n less than 550?

(1) The product of the digits in n is 30 --> $$30=1*2*3*5$$ --> n can be any combination of digits from the following two sets: {1, 6(=2*3), 5} or {2, 3, 5}. n can be lees than 550 (eg. 165, 156, ...) OR more than 550 (eg 615, 651). Not sufficient.

(2) The sum of the digits in n is 10 --> Clearly not sufficient. Many combinations are possible 109, 901, ....

(1)+(2) We have the set {2, 3, 5} (as only in this set the sum of the digits is 10). From this set we can not form three digit number more than 550 (max=532). Sufficient.

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Re: Is the three-digit number n less than 550?  [#permalink]

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30 Nov 2013, 00:10
Its C and not E. A lil tricky if u don't pay careful attention
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Re: Is the three-digit number n less than 550?  [#permalink]

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18 Dec 2016, 12:53
Thanks alot, the explanation was very helpful. I always struggle with that kind of questions, do you have any recommendations?
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Re: Is the three-digit number n less than 550?  [#permalink]

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18 Feb 2017, 03:00
Is the three-digit number n less than 550?

Statement 1 is not sufficient because N can be:

561 obviously greater than 550
or
235 obviously less than 550

Statement 2 is not sufficient because it can be

109 obviously less than 550
or
901 obviously greater than 550

Together
N can be 235 or 532 both of which is less than 550.

Hence C.
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Re: Is the three-digit number n less than 550?  [#permalink]

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07 Nov 2018, 17:56
Stiv 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.

$$100 \leqslant \,\,n = \left\langle {ABC} \right\rangle \,\, \leqslant 999$$

$$\left\langle {ABC} \right\rangle \,\,\mathop < \limits^? \,\,550$$

$$\left( 1 \right)\,\,\,A \cdot B \cdot C = 30\,\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,\left( {A,B,C} \right) = \left( {1,5,6} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\ \,{\text{Take}}\,\,\left( {A,B,C} \right) = \left( {5,6,1} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\ \end{gathered} \right.$$

$$\left( 2 \right)\,\,\,A + B + C = 10\,\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,\left( {A,B,C} \right) = \left( {1,2,7} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\ \,{\text{Take}}\,\,\left( {A,B,C} \right) = \left( {5,5,0} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\ \end{gathered} \right.$$

$$\left( {1 + 2} \right)\,\,\,\left\langle {ABC} \right\rangle \,\,\, \geqslant \,\,\,550\,\,\,\, \Rightarrow \,\,\,\left\{ \begin{gathered} \,A = 5\,\,,\,\,B \geqslant 5\,\,\,\mathop \Rightarrow \limits^{\left( 2 \right)} \,\,\,B = 5,\,\,C = 0\,\,\, \Rightarrow \,\,\,\left( 1 \right)\,\,{\text{contradicted}} \hfill \\ \,\,\,{\text{or}} \hfill \\ \,A \geqslant 6\,\,\,\mathop \Rightarrow \limits_{\left( 1 \right)}^{\left( * \right)} \,\,\,A = 6\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,B \cdot C = 5\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,A + B + C = 12\,\,\, \Rightarrow \,\,\,\left( 2 \right)\,\,{\text{contradicted}} \hfill \\ \end{gathered} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle$$

$$\left( * \right)\,\,\,\left( 1 \right)\,\,\, \Rightarrow \,\,\,A,B,C\,\,{\text{are}}\,\,\left( {{\text{positive}}} \right)\,\,{\text{divisors}}\,\,{\text{of}}\,\,30\,\,\, \Rightarrow \,\,\,A \notin \left\{ {7,8,9} \right\}$$

$$\left( {**} \right)\,\,\,B \cdot C = 5\,\,\, \Rightarrow \,\,\,B,C\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{pair}}\,\,{\text{of}}\,\,\left( {{\text{positive}}} \right)\,\,{\text{divisors}}\,\,{\text{of}}\,\,5\,\,\, \Rightarrow \,\,\,\left\{ \begin{gathered} \,\left( {B,C} \right) = \left( {1,5} \right) \hfill \\ \,\,{\text{or}} \hfill \\ \,\left( {B,C} \right) = \left( {5,1} \right) \hfill \\ \end{gathered} \right.$$

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

Regards,
Fabio.
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Re: Is the three-digit number n less than 550? &nbs [#permalink] 07 Nov 2018, 17:56
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