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October 29, 2019

Combinatorics, Israel MO 2014, Problem 4

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Understand the problem

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The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?
[/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="3.22.4" 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="4.0" hover_enabled="0"]Israel 2014, Problem 4 [/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="4.0" hover_enabled="0" open="off"]Combinatorics, Number Theory [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.0" hover_enabled="0" open="off"]6/10 [/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.0" hover_enabled="0" open="off"]Excursion in Mathematics by Bhaskarcharya Prathistan [/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"]

Start with hints

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[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="4.0" hover_enabled="0"]Let's write the problem mathematically, i.e. in terms of the equations. Let's write the condition mathematically. Let $abcd$ be a four-digit number, with $1\le a\le9$ and $0\le b,c,d\le 9$, and $a,b,c,d$ positive integers. Then we need to have $1000a+100b+10c+d=27n$ and $a+b+c+d=27m$, where $n,m$ are positive integers.  Now, we have to count the number of such solutions. Observe that the sum of the digits can be at most 36. So, m = 1. Hence, a+b+c+d = 27. This leads to $111a+11b+c=3(n-1)$ . This implies  $2b+c$ being a multiple of $3$.  Now, this implies we have quite a number of cases to investigate. Let's do them one by one patiently. [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="4.0" hover_enabled="0"]
bca+d
03,6,927,24,21,18
Out of this b = 0, c = 9, a+d = 18, a = 9, c =9 is the only possibility considering the maximum possibble value of a+d being 18.  
bca+d
11,4,725,22,19
  None of the cases is fine as the maximum possible value of a+d is 18. [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="4.0" hover_enabled="0"]
bca+d
22,5,823,20,17
This gives rise to b = 2, c = 8, a+d = 17. Hence two solutions 8289 and 9288.
bca+d
30,3,6,924,21,18,15
Thus you can understand this gives rise to 4 solutions: ($6399,7398,8397$ and $9396$) [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="4.0" hover_enabled="0"]
bca+d
41,4,722,19,16
This gives rise to three solutions: ($7479,8478,$ and $9477$).
bca+d
52,5,820,17,14
Only the last two choices are acceptable; the former gives us two solutions, and the latter 5 (for a total of seven solutions).   [/et_pb_tab][et_pb_tab title="HInt 5" _builder_version="4.0" hover_enabled="0"]
bca+d
60,3,6,921,18,15,12
 The last three choices are acceptable and give us $1+4+7=12$ solutions.
bca+d
71,4,719,16,13
The last two choices are acceptable and give us $3+6=9$ solutions.
bca+d
82,5,817,14,11
All choices are acceptable and give us $2+5+8=15$ solutions.
bca+d
90,3,6,918,15,15,9
All choices are acceptable and give us $1+4+7+9=21$ solutions. Therefore, there are $1+2+5+3+7+12+9+15+21=75$ four-digit numbers with this property. [/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"]

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Similar Problems

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