Difference between revisions of "1982 AHSME Problems/Problem 30"
MRENTHUSIASM (talk | contribs) (Created page with "== Problem == Find the units digit of the decimal expansion of <cmath>\left(15 + \sqrt{220}\right)^{19} + \left(15 + \sqrt{220}\right)^{82}.</cmath> <math>\textbf{(A)}\ 0\qqu...") |
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== Solution == | == Solution == | ||
| + | Let <math>A=15+\sqrt{220}</math> and <math>B=15-\sqrt{220}.</math> Note that <math>A^{19}+B^{19}</math> and <math>A^{82}+B^{82}</math> are both integers: When we expand (Binomial Theorem) and combine like terms for each expression, the rational terms are added and the irrational terms are canceled. We have | ||
| + | <cmath>\begin{align*} | ||
| + | A^{19}+B^{19} &= \left[\binom{19}{0}15^{19}\sqrt{220}^0+\binom{19}{1}15^{18}\sqrt{220}^1+\cdots+\binom{19}{19}15^0\sqrt{220}^{19}\right] + \left[\binom{19}{0}15^{19}\sqrt{220}^0-\binom{19}{1}15^{18}\sqrt{220}^1+\cdots-\binom{19}{19}15^0\sqrt{220}^{19}\right] \\ | ||
| + | &= 2\left[\binom{19}{0}15^{19}\sqrt{220}^0+\binom{19}{2}15^{17}\sqrt{220}^2+\cdots+\binom{19}{18}15^1\sqrt{220}^{18}\right]. | ||
| + | \end{align*}</cmath> | ||
== See Also == | == See Also == | ||
{{AHSME box|year=1982|num-b=29|after=Last Problem}} | {{AHSME box|year=1982|num-b=29|after=Last Problem}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 22:35, 11 September 2021
Problem
Find the units digit of the decimal expansion of ![\[\left(15 + \sqrt{220}\right)^{19} + \left(15 + \sqrt{220}\right)^{82}.\]](http://latex.artofproblemsolving.com/3/7/2/37278df376055897e1132e53b5a6d5bb9a92c3b1.png)
Solution
Let 
and 
Note that 
and 
are both integers: When we expand (Binomial Theorem) and combine like terms for each expression, the rational terms are added and the irrational terms are canceled. We have
![\begin{align*} A^{19}+B^{19} &= \left[\binom{19}{0}15^{19}\sqrt{220}^0+\binom{19}{1}15^{18}\sqrt{220}^1+\cdots+\binom{19}{19}15^0\sqrt{220}^{19}\right] + \left[\binom{19}{0}15^{19}\sqrt{220}^0-\binom{19}{1}15^{18}\sqrt{220}^1+\cdots-\binom{19}{19}15^0\sqrt{220}^{19}\right] \\ &= 2\left[\binom{19}{0}15^{19}\sqrt{220}^0+\binom{19}{2}15^{17}\sqrt{220}^2+\cdots+\binom{19}{18}15^1\sqrt{220}^{18}\right]. \end{align*}](http://latex.artofproblemsolving.com/b/3/a/b3aa098271edd1706bb8fe406b4bc31572cb2ae1.png)
See Also
| 1982 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 29 |
Followed by Last Problem | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
