, where the second factor has 
digits, is an integer whose digits have a sum of 
. What is ?[/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="4.0" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="off" _builder_version="4.0"]American Mathematics Competition [/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"]7/10[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.29.2" open="on"]Elementary Number Theory by David M, Burton
[/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"]So, well have a long look at the problem. With a little bit of thought, you might even crack this without proceeding any further !
[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="4.0"]Think you could do with some help ? Okay let's get ourselves a headstart. As you might have guessed, sometimes the most fruitful thing to be done is to observe what's going on in these kind of problems. The intuition is clear, there are too many instances of '8'-s for someone to account for them manually. So, yes, that's something we can expect. Try working out with a few simple cases like, k=1, k=2, and so on...
[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="4.0"]
Okay, so let's see what's happening for a few small values of k...
k=1
8 * 8 = 64
k=2
8 * 88 = 704
k=3
8 * 888 = 7104
k=4
8 * 8888 = 71104
...
So, Wait ! Look closely...Do you see something ?
A really nice pattern is evolving. If you can see it, and yet find it a bit difficult to articulate it mathematically, don't worry. See for k=4, the product gives us the result 71104. That means, we have the starting digit to be 7. And the ending suffix is 04. What varies are the ones. See, for k=4, we have ( k-2 = 2 ) ones. That's it ! The generalization should be fairly simple for us to do now... For every k >=2, the product result consists of a 7 to start with, exactly k-2 1's follow, and we conclude with single occurrences of 0 and 4 each. Now, think...can you take this till the end ?
[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="4.0"]Once we've seen the pattern, it's easy to get this done.
What we have noted, we need that to sum up to 1000.
In simple mathematical terms,
7 + (k-2).1 + 0 + 4 = 1000
11 + k = 1002
k = 991
So, we need '8' to come 991 times in the multiplicand, so that the digits sum up to 1000. So, that seals the deal !
[/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"]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://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"]
