Awesome Algebra

Algebra Level pending

If a times a plus b times b equals c times c, and a,b, and c are positive integers, what is the lowest possible value of c?


The answer is 5.

Navpreet Singh by Navpreet Singh , 42, USA

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3 solutions

Alex Zhong
Apr 8, 2015

a 2 + b 2 = c 2 a^2+b^2=c^2 fits Pythagorean triplets: (3,4,5)

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Caleb Townsend
Apr 8, 2015

a 2 + b 2 = c 2 a , b , c ∈ Z + a^2 + b^2 = c^2 \\ a,\ b,\ c \in \mathbb{Z^+} So we are looking for the square root smallest square that can be written as the sum of two positive squares. Simply checking the first five perfect squares gives the answer.

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Or you can just note the parametric equation for c c is m 2 + n 2 m^2+n^2 for positive integers m ≠ n m \ne n , with the minimum value occur at m = 1 , n = 2 ⇒ c = 5 m=1,n=2 \Rightarrow c = 5

Pi Han Goh - 6 years, 2 months ago
Navpreet Singh
Apr 8, 2015

The answer is 5;. This is because a=3 and b=4. a times a equals 9. b times b equals 16. 16 plus 9 equals 25. The square root of 25 is 5. c=5.

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