Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

Divisibility Problem- AMC 8 (2016) - Question 5

[et_pb_section fb_built="1" _builder_version="4.2.2"][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="4.2.2" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Aclonica|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" inline_fonts="Aclonica"]

What are we learning ?

[/et_pb_text][et_pb_text _builder_version="4.2.2" text_font="Raleway||||||||" text_font_size="20px" text_letter_spacing="1px" text_line_height="1.5em" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]

Competency in Focus: Divisibility.

This problem from American Mathematics contest (AMC 8, 2016) is based on the concept of divisibility .

[/et_pb_text][et_pb_text _builder_version="4.2.2" text_font="Aclonica|300|||||||" text_text_color="#ffffff" header_font="Aclonica|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" inline_fonts="Aclonica"]

First look at the knowledge graph.

[/et_pb_text][et_pb_image src="https://www.cheenta.com/wp-content/uploads/2020/02/amc8_2016_5.png" align="center" force_fullwidth="on" _builder_version="4.2.2" min_height="388px" height="198px" max_height="207px"][/et_pb_image][et_pb_text _builder_version="4.2.2" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" header_2_font="Aclonica||||||||" background_color="#0c71c3" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" inline_fonts="Aclonica"]

Next understand the problem

[/et_pb_text][et_pb_text _builder_version="4.2.2" text_font="Raleway||||||||" text_font_size="20px" text_letter_spacing="1px" text_line_height="1.5em" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]The number $N$ is a two-digit number. • When $N$ is divided by $9$, the remainder is $1$. • When $N$ is divided by $10$, the remainder is $3$. What is the remainder when $N$ is divided by $11$?   $\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }7$[/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="4.2.2" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" hover_enabled="0"][et_pb_accordion_item title="Source of the problem" _builder_version="4.2.2" open="off"]American Mathematical Contest 2016, AMC 8  Problem 5[/et_pb_accordion_item][et_pb_accordion_item title="Key Competency" _builder_version="4.2.2" hover_enabled="0" open="off"]

Divisibility

[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.2.2" open="off"]4/10[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" open="off" _builder_version="4.2.2" hover_enabled="0"]

Challenge and thrill of Pre-College mathematics [/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="4.0.9" 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" inline_fonts="Aclonica"]

Start with hints 

[/et_pb_text][et_pb_tabs _builder_version="4.2.2"][et_pb_tab title="HINT 0" _builder_version="4.0.9"]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title="HINT 1" _builder_version="4.2.2"]When $N$ is divided by $10$ it leaves remainder $3$ i.e., $N=10\times P+3 \textbf{ ,where } P$ is an integer. i.e., The unit digit of $N$ must be $3$ because unit digit of $10P$ is zero.[/et_pb_tab][et_pb_tab title="HINT 2" _builder_version="4.2.2"]$N$ leaves remainder $1$ when divided by $9$. i.e., $N=9\times Q+1$ where $Q$ is an integer. Since $10\times P+3=9\times Q+1$ the unit digit of $9\times Q +1 $ must be $3$.        [/et_pb_tab][et_pb_tab title="HINT 3" _builder_version="4.2.2"]Since $N$ is a two digit number then the only possibility is $Q=8$ i.e., $N=9\times 8+1=73$[/et_pb_tab][/et_pb_tabs][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section fb_built="1" fullwidth="on" _builder_version="4.2.2" global_module="50833"][et_pb_fullwidth_header title="AMC - AIME Program" button_one_text="Learn More" button_one_url="https://www.cheenta.com/amc-aime-usamo-math-olympiad-program/" header_image_url="https://www.cheenta.com/wp-content/uploads/2018/03/matholympiad.png" _builder_version="4.2.2" title_level="h2" background_color="#00457a" custom_button_one="on" button_one_text_color="#44580e" button_one_bg_color="#ffffff" button_one_border_color="#ffffff" button_one_border_radius="5px"]

AMC - AIME - USAMO Boot Camp for brilliant students. Use our exclusive one-on-one plus group class system to prepare for Math Olympiad

[/et_pb_fullwidth_header][/et_pb_section][et_pb_section fb_built="1" fullwidth="on" _builder_version="4.2.2" global_module="50840"][et_pb_fullwidth_post_slider include_categories="879,878,869" show_arrows="off" show_pagination="off" show_meta="off" image_placement="left" _builder_version="4.2.2" custom_button="on" button_text_color="#0c71c3" button_bg_color="#ffffff" custom_margin="20px||20px||false|false" custom_padding="20px||20px||false|false"][/et_pb_fullwidth_post_slider][/et_pb_section]

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com