Question 51
Let $n$ be a natural number such that $n = a^{2} + b^{2} + c^{2}$, for some natural numbers $a, b, c$. Prove that
$$
9 n = (p_{1} a + q_{1} b + r_{1} c)^{2} + (p_{2} a + q_{2} b + r_{2} c)^{2} + (p_{3} a + q_{3} b + r_{3} c)^{2}
$$
where $p_{j}$'s, $q_{j}$'s, $r_{j}$'s are all nonzero integers. Further, if $3$ does not divide at least one of $a, b, c$, prove that $9 n$ can be expressed in the form $x^{2} + y^{2} + z^{2}$, where $x, y, z$ are natural numbers none of which is divisible by $3$.
$$
9 n = (p_{1} a + q_{1} b + r_{1} c)^{2} + (p_{2} a + q_{2} b + r_{2} c)^{2} + (p_{3} a + q_{3} b + r_{3} c)^{2}
$$
where $p_{j}$'s, $q_{j}$'s, $r_{j}$'s are all nonzero integers. Further, if $3$ does not divide at least one of $a, b, c$, prove that $9 n$ can be expressed in the form $x^{2} + y^{2} + z^{2}$, where $x, y, z$ are natural numbers none of which is divisible by $3$.