Which of the following is/are perfect numbers if "Perfe

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

by GMATinsight » Mon Oct 08, 2018 12:18 am

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Which of the following is/are perfect numbers if "Perfect Number" is defined as number for which sum of all it's factors is twice the number.
I) 6
II) 28
III) 36

A) I only
B) II only
C) III only
D) I and II only
D) I, II and III

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Source: Beat The GMAT — Problem Solving | Mon Oct 08, 2018 12:18 am

fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

Which of the following is/are perfect numbers if "Perfe

by fskilnik@GMATH » Mon Oct 08, 2018 5:01 am

GMATinsight wrote:Which of the following is/are perfect numbers? A "perfect number" is defined as an integer such that the sum of all its POSITIVE factors is twice the integer itself.

I) 6
II) 28
III) 36

A) I only
B) II only
C) III only
D) I and II only
E) I, II and III

$$?\,\,\,:\,\,\,{\rm{perfect}}\,\,{\rm{number}}$$
$${\rm{6}} \to {\rm{1 + 6 + 2 + 3}}\,\,\mathop {\rm{ = }}\limits^? \,\,{\rm{2}} \cdot {\rm{6}}\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\, \Rightarrow \,\,\,\,\,{\rm{refute}}\,\,\,\left( B \right)\,\,,\,\,\left( C \right)\,\,$$
$${\text{28}} \to {\text{1 + 28 + 2 + 14 + 4 + 7}}\,\,\mathop {\text{ = }}\limits^? \,\,2 \cdot 28\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\, \Rightarrow \,\,\,\,\left( {{\text{and}}} \right)\,\,\,{\text{refute }}\left( A \right)\,\,$$
$$36 \to 1 + 36 + 2 + 18 + 3 + 12 + 4 + 9 + 6\,\,\mathop {\text{ = }}\limits^? \,\,2 \cdot 36\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\, \Rightarrow \,\,\boxed{\,\,{\text{?}}\,\,\,{\text{ = }}\,\,\,\left( D \right)\,\,}{\text{ }}$$

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

Regards,
Fabio.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8136; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Wed Oct 10, 2018 5:49 pm

GMATinsight wrote:Which of the following is/are perfect numbers if "Perfect Number" is defined as number for which sum of all it's factors is twice the number.
I) 6
II) 28
III) 36

A) I only
B) II only
C) III only
D) I and II only
D) I, II and III

The sum of the factors of 6 are 1 + 2 + 3 + 6 = 12, which is 2 x 6. So 6 is a perfect number.

The sum of the factors of 28 are 1 + 2 + 4 + 7 + 14 + 28 = 56, which is 2 x 28. So 28 is a perfect number.

The sum of the factors of 36 are 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 36 = 91, which is not 2 x 36. So 36 is not a perfect number.

Answer: D

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