Question

Find the units digit in the decimal expansion of $(15+\sqrt{220}{)}^{19}+(15+\sqrt{220}{)}^{82}$.

• A. $0$
• B. $2$
• C. $5$
• D. $9$
• E. None of these

Answer 1

Answer: (D)

$9$

We can see by binomial expansion that $(15+\sqrt{220}{)}^{19}+(15-\sqrt{220}{)}^{19}+(15+\sqrt{220}{)}^{82}+(15-\sqrt{220}{)}^{82}$

is a multiple of $10$.

We can also see that $0<15-\sqrt{220}<0.5$ since $15-\sqrt{220}=\frac{5}{15+\sqrt{220}}$

So, $0<(15-\sqrt{220}{)}^{19}+(15-\sqrt{220}{)}^{82}<1$

So, $(15+\sqrt{220}{)}^{19}+(15+\sqrt{220}{)}^{82}$ is a multiple of $10$ minus a positive real number that is less than $1$.

In particular, its integer part must end in the digit $9$.

So, option D is correct.