INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

May 19, 2019

Product of Digits, ISI Entrance 2017

[et_pb_section fb_built="1" _builder_version="3.22.4"][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="4.3.4" 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" inline_fonts="Aclonica"]

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"]  Let \(g : \mathbb{N} \to \mathbb{N} \) with \( g(n) \) being the product of digits of \(n\).        (a) Prove that \( g(n)\le n\) for all \( n \in \mathbb{N} \) .        (b) Find all \(n \in \mathbb{N} \) , for which \( n^2-12n+36=g(n) \). [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25" link_option_url="https://www.cheenta.com/isicmientrance/" link_option_url_new_window="on"][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="4.3.4" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" hover_enabled="0"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.3.4" hover_enabled="0"]
I.S.I. (Indian Statistical Institute, B.Stat, B.Math) Entrance. Subjective Problem 5 from 2017
[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.22.4" open="off"]Inequality

[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.22.4" open="off"]9 out of 10

[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.3.4" hover_enabled="0" open="off"]

Challenge and Thrill of Pre-College Mathematics by V.Krishnamuthy , C.R.Pranesachar, ect. [/et_pb_accordion_item][/et_pb_accordion][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" inline_fonts="Aclonica"]

Start with hints

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.22.4" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" custom_padding="|||12px||"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]

Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.22.4"]

Let \(n\) be a \(k\) digit(s) number number , then \(n\) can be written as                             \(n=a_0+10a_1+10^2a_2+\cdots+10^{k-1}a_{k-1}\) Where ,\(a_0,a_1,...,a_{k-1}\) are digits of \(n\).

[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.22.4"]

\(a_0,a_1,...,a_{k-1} \in [1,9] \) as the range of the function \(g\) is \(\mathbb{N}\) \(\Rightarrow a_0,a_1,...,a_{k-1}\neq 0\) . Now \(g(n)=a_0a_1a_2\cdots a_{k-1}\le \underbrace{10\cdot10\cdot10\cdots 10}_{(k-1) times}  \cdot a_{k-1}\)   [Since \(a_0,a_1,...,a_{k-1} \le 10 \)]  Equality holds when \(k=1\) .

[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.22.4"]

  \(\Rightarrow g(n)\le 10^{k-1}\cdot a_{k-1}\) \(\Rightarrow g(n)\le 10^{k-1}\cdot a_{k-1}+\cdots+10^2a_2+10a_1+a_0\)    [Since \(a_0,a_1,...,a_{k-1} >0\)]  \(\Rightarrow g(n)\le n\)        (Proved) .  

[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.22.4"]

\(n^2-12n+36=g(n)\) \(\Rightarrow n^2-12n+36 \le n\)     [Since \(g(n) \le n\) ] \(\Rightarrow n^2-13n+36 \le 0\) \(\Rightarrow (n-9)(n-4) \le 0\) \(\Rightarrow 4\le n\le 9\) \(\Rightarrow n={4,5,6,7,8,9}\) (Ans.)  . 

[/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" min_height="12px" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" inline_fonts="Aclonica"]

Connected Program at Cheenta

[/et_pb_text][et_pb_blurb title="I.S.I. & C.M.I. Entrance Program" image="https://www.cheenta.com/wp-content/uploads/2018/03/ISI.png" _builder_version="3.22.4" header_level="h1" header_font="||||||||" header_text_color="#e02b20" header_font_size="50px" body_font="||||||||"]

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

[/et_pb_blurb][et_pb_button button_url="https://www.cheenta.com/matholympiad/" button_text="Learn More" button_alignment="center" _builder_version="3.22.4" custom_button="on" button_text_color="#ffffff" button_bg_color="#e02b20" 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" inline_fonts="Aclonica"]

Similar Problems

[/et_pb_text][et_pb_post_slider include_categories="9" _builder_version="3.22.4"][/et_pb_post_slider][et_pb_divider _builder_version="3.22.4" background_color="#0c71c3"][/et_pb_divider][/et_pb_column][/et_pb_row][/et_pb_section]

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

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