Indeterminate Equation #2

Let $M=(10p+q)^3=(p+q+3)^5$ .

Then we can see that $M$ is a fifteenth power of a natural number, which makes $(10p+q)$ a fifth power of that natural number.

But since $(10p+q)$ is a two-digit number, it must be $2^5=32$ .

$\therefore p=3,\quad q=2$

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Let $N=(100r+10s+t)^2=(r+s+t)^5$ .

Then we can see that $N$ is a tenth power of a natural number, which makes $(100r+10s+t)$ a fifth power of that natural number.

But since $(100r+10s+t)$ is a three-digit number, it must be $3^5=243$ .

$\therefore r=2,\quad s=4,\quad t=3$

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$\therefore p^2+q^2+r^2+s^2+t^2=3^2+2^2+2^2+4^2+3^2=\boxed{42}$