[et_pb_section fb_built="1" _builder_version="4.0"][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]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
[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]Start with hints
[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||48px" custom_padding="20px|20px|0px|20px||" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]Watch video
[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" min_height="12px" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]Connected Program at Cheenta
[/et_pb_text][et_pb_blurb title="Amc Training Camp" url="https://cheenta.com/matholympiad/" url_new_window="on" image="https://cheenta.com/wp-content/uploads/2018/03/matholympiad.png" _builder_version="3.23.3" header_font="||||||||" header_text_color="#e02b20" header_font_size="48px" link_option_url="https://cheenta.com/matholympiad/" link_option_url_new_window="on"]Cheenta AMC Training Camp consists of live group and one on one classes, 24/7 doubt clearing and continuous problem solving streams.
Start for free.
[/et_pb_blurb][et_pb_button button_url="https://cheenta.com/matholympiad/" url_new_window="on" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" background_layout="dark" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3"][/et_pb_button][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]Similar Problems
[/et_pb_text][et_pb_divider _builder_version="3.22.4" background_color="#0c71c3"][/et_pb_divider][/et_pb_column][/et_pb_row][et_pb_row _builder_version="4.0"][et_pb_column type="4_4" _builder_version="4.0"][et_pb_post_nav in_same_term="on" _builder_version="4.0"][/et_pb_post_nav][/et_pb_column][/et_pb_row][/et_pb_section]
