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## Competency in Focus: Divisibility.

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

## 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″ hover_enabled=”0″ open=”on”]

American Mathematical Contest 2016, AMC 8  Problem 5 [/et_pb_accordion_item][et_pb_accordion_item title=”Key Competency” open=”off” _builder_version=”4.2.2″]

### Divisibility

[/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]
Categories

## Competency in Focus: Divisibility.

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

## 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″]

[/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”]