Old is gold

Number Theory Level 3

Find the number of non-negative integral solutions of the equation given below.

a 3 + b 3 + c 3 = 1969 \sqrt{a^3+b^3+c^3}=1969

This problem is not original.
4 3 none of them 5 0 16 9 6

Atul Shivam by Atul Shivam , 23, India

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1 solution

Otto Bretscher
Jan 7, 2016

The cube of any positive integer is congruent to 0, 1, or -1 mod 9, so that the sum of three cubes cannot be congruent to 4 or 5. Since 196 9 2 ≡ 4 ( m o d 9 ) 1969^2\equiv 4\pmod9 , the given equation has 0 \boxed{0} solutions.

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