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Difference between revisions of "2024 SSMO Team Round Problems/Problem 11"

(Created page with "==Problem== Let <math>S</math> denote the set of positive divisors of <math>5400.</math> Let <cmath>S_i = \{d \mid d \in S, \, d \equiv i \Mod4\}</cmath> and let <math>s_i</m...")
 
 
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==Problem==
 
==Problem==
  
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Let <math>S</math> denote the set of positive divisors of <math>5400.</math> Let <cmath>S_i = \{d \mid d \in S, \, d \equiv i \Mod4\}</cmath> and let <math>s_i</math> denote the sum of all elements of <math>S_i.</math> Find the value of <cmath>s_0^2+s_1^2+s_2^2+s_3^2-2s_0s_2-2s_1s_3.</cmath>
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Let <math>S</math> denote the set of positive divisors of <math>5400.</math> Let <cmath>S_i = \{d \mid d \in S, \, d \equiv i \pmod4\}</cmath> and let <math>s_i</math> denote the sum of all elements of <math>S_i.</math> Find the value of <cmath>s_0^2+s_1^2+s_2^2+s_3^2-2s_0s_2-2s_1s_3.</cmath>
  
 
==Solution==
 
==Solution==
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Note that <math>5400 = 2^3\cdot3^3\cdot5^2.</math> Firstly, we have <cmath>s_0^2+s_1^2+s_2^2+s_3^2-2s_0s_2-2s_1s_3=(s_0-s_2)^2+(s_1-s_3)^2.</cmath> Now, let <math>s(n)</math> denote the sum of positive divisors of <math>n.</math> We have <cmath>s_0-s_2 = (8+4-2)s(3^3\cdot5^2) = 10s(3^3\cdot5^2) = 10s(3^3)s(5^2) = 400s(5^2).</cmath> In addition, <cmath>s_3-s_1 = (27-9+3-1)s(5^2) = 20s(5^2).</cmath> It is easy to compute <math>s(5^2) = 31.</math> So, our answer is <math>420s(5^2) = \boxed{13020}.</math>
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~SMO_Team

Latest revision as of 14:41, 10 September 2025

Problem

Let $S$ denote the set of positive divisors of $5400.$ Let \[S_i = \{d \mid d \in S, \, d \equiv i \pmod4\}\] and let $s_i$ denote the sum of all elements of $S_i.$ Find the value of \[s_0^2+s_1^2+s_2^2+s_3^2-2s_0s_2-2s_1s_3.\]

Solution

Note that $5400 = 2^3\cdot3^3\cdot5^2.$ Firstly, we have \[s_0^2+s_1^2+s_2^2+s_3^2-2s_0s_2-2s_1s_3=(s_0-s_2)^2+(s_1-s_3)^2.\] Now, let $s(n)$ denote the sum of positive divisors of $n.$ We have \[s_0-s_2 = (8+4-2)s(3^3\cdot5^2) = 10s(3^3\cdot5^2) = 10s(3^3)s(5^2) = 400s(5^2).\] In addition, \[s_3-s_1 = (27-9+3-1)s(5^2) = 20s(5^2).\] It is easy to compute $s(5^2) = 31.$ So, our answer is $420s(5^2) = \boxed{13020}.$

~SMO_Team

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