# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]For how many positive integers $n$ does 1+2+3+4+....+n evenly divide from 6n? (a)3.       (b)5.       (c)7.       (d)9.       (e)11

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="4.0"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="4.0" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.0"]American Mathematical Contest 2005 10A Problem 21

[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="4.0" open="off"]Number Theory

[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.0" open="off"]6/10

[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.0" open="off"]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics

[/et_pb_accordion_item][/et_pb_accordion][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="4.0" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" min_height="148px" custom_padding="||24px|20px||"][et_pb_tab title="Hint 0" _builder_version="4.0"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="4.0"]Step 1. So after having a deep look into this problem you can see that if 1+2+3+.....+n evenly divides 6n that is $\frac{6n}{1+2+3+....+n}$ now to think about formula of the sum of 1+2+3+.....+n.

[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="4.0"]

Step 2. After getting the formula as 1+2+3+4+....+n=$\frac{n(n+1)}{2}$ substitute it in the equation $\frac{6n}{1+2+3+....+n}$ and simplify it. Give it a try!!!!!!

[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="4.0"]

Step 3 Now by simplifying you will get $\frac{12}{n+1}$. Now here lies the main concept of this problem as you have to find integer n so you must see that if (n+1) is a factor of 12 then only $\frac{12}{n+1}$ will become an integer. Now find out the factors of 12 and try to build up some logic how to make this $\frac{12}{n+1}$ an integer.

[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="4.0"]

Step 4 So you can easily say that the factors of 12 are 1,2,3,4,6 and 12 respectively now try to think who you can use this information here in this $\frac{12}{n+1}$. Like what are the values of n (from the factors of 12) in order to make it a (n+1) factor of 12.

[/et_pb_tab][et_pb_tab title="Hint 5" _builder_version="4.0"]

Step 5 . Here n can take values 0,1,2,3,5 and 11 respectively as n+1 must be a factor of  12 . But here 0 is not a positive integer so you have to exclude 0 so you are left with 5 different values of n . So your answer is 5

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