The units digit in the integral part of the expansion of (15+220)19(1+(15+220)63) is ___ (Here integral part means the greatest integer less than or equal to that number)

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The units digit in the integral part of the expansion of ${(15 + \sqrt{220})}^{19} (1 + {(15 + \sqrt{220})}^{63})$ is ___
(Here integral part means the greatest integer less than or equal to that number)

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answer is 9.

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Detailed Solution

Let ${(15 + \sqrt{220})}^{19} = I + f, 0 < f < 1$ $\Rightarrow {(15 - \sqrt{220})}^{19} = f_{1}, 0 < f_{1} < 1$
Adding both, $I + 1 = 10 λ$
${(15 + \sqrt{220})}^{82} = I_{1} + f_{2}, 0 < f_{2} < 1 \Rightarrow {(15 - \sqrt{220})}^{82} = f_{3}, 0 < f_{3} < 1$
Adding both, $I_{1} + 1 = 10 λ_{1}$
So, the value of $I + I_{1}$ will be of the type $10 λ_{3} - 2$
But, we also get $f + f_{2} > 1$