Let a, b, c be three distinct positive integers such that the sum of any two of them is a
perfect square and having minimal sum, a + b + c. Find thesum.
Answers
Given : a,b,c be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum, a + b + c
To Find : a + b + c
Solution:
a + b = k²
a + c = m²
b + c = n²
=> b - c = k² - m²
=> b + c = n²
=> 2b = k² - m² + n²
=> b = (k² - m² + n²)/2
a = (k² + m² - n²)/2
c = ( -k² + m² + n²)/2
=>k² , m² , n² all 3 has to be even or 2 odd and 1 even
and sum of any two > third as a , b & c are positive
with some hit and trial Earliest possible combination is
k² = 25
m² = 36
n² = 49
Hence a , b & c are
6 , 19 , 30
a + b + c = 6 + 19 + 30 = 45
minimal sum, a + b + c. = 45
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Given :-
a,b,c be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum, a + b + c
To Find : a + b + c
Solution:-
a + b = k²
a + c = m²
b + c = n²
=> b - c = k² - m²
=> b + c = n²
=> 2b = k² - m² + n²
=> b = (k² - m² + n²)/2
a = (k² + m² - n²)/2
c = ( -k² + m² + n²)/2
=>k² , m² , n² all 3 has to be even or 2 odd and 1 even
and sum of any two > third as a , b & c are positive
with some hit and trial Earliest possible combination is
k² = 25
m² = 36
n² = 49
Hence a , b & c are
6 , 19 , 30
a + b + c = 6 + 19 + 30 = 45
minimal sum, a + b + c. = 45