Find all positive integers $n$ such that: $$n = a^2 + b^2 + c^2 + d^2,$$where $a < b < c < d$ are the smallest divisors of $n$.
Problem
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Tags: number theory
miyukina
24.03.2024 09:48
a = 1
n cannot be odd from this summing, so n must have at least a factor of 2
===> b = 2
Because n is even then c and d must be a pair of even-odd numbers
With the sum of two even and two odd squares, n = 2 mod 4, so it must be that c is odd and d = 2c
n = 1^2 + 2^2 + c^2 + (2c)^2
= 5 × (1 + c^2)
Since 5 | n, then c = 5 only
Answer, n
= 5 × (1^2 + 5^2)
= 130
SomeonecoolLovesMaths
24.03.2024 10:10
here you go
megarnie
24.03.2024 21:10
https://artofproblemsolving.com/community/c1975135h2858365p25365289