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August 12, 2019

Missing digits of 34!

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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"]34!=295232799cd96041408476186096435ab000000 Find a,b,c,d (all single digits).

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][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="3.26.6" 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="3.26.6"]

BMO 2002[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.26.6" open="off"]Number Theory[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.26.6" open="off"]Easy[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.26.6" open="off"]An Excursion in Mathematics[/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

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[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.26.6"]Get prepared to find the residue of 34! modulo various divisors! The substitution x=\frac{34!}{10^7}=<span><span class="mrow" id="MathJax-Span-300"><span class="mn" id="MathJax-Span-304">295232799</span><span class="mi" id="MathJax-Span-305">c</span><span class="mi" id="MathJax-Span-306">d</span><span class="mn" id="MathJax-Span-307">9604140809643</span><span class="mi" id="MathJax-Span-308">a</span><span class="mi" id="MathJax-Span-309">b should help simplify the calculation.
[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.26.6"]Note that x is divisible by 2^7. Find all possible residues of a number modulo 10^7 given that the number is divisible by 2^7. That'll help you prove that a=5, b=2.[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.26.6"]Note that x is divisible by 9. This gives that c+d is either 3 or 12.[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.26.6"]As x is also divisible by 11, we must have c-d=-3 or 8. As 2c has to be an even integer less than or equal to 18, we must have c+d=3, c-d=-3. This gives c=0, d=3.[/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" inline_fonts="Aclonica"]

Watch video

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Connected Program at Cheenta

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Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.[/et_pb_blurb][et_pb_button button_url="https://www.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" inline_fonts="Aclonica"]

Similar Problems

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