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1990 AMC 12 Problem 30

Attempt Problem 30 of the 1990 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1990 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

30.

If Rn=12(an+bn),R_n=\frac12(a^n+b^n), where a=3+22,a=3+2\sqrt2, b=3−22,b=3-2\sqrt2, and n=0,n=0, 1,1, 2,2, …,\ldots, then R12345R_{12345} is an integer. Its units digit is

11

33

55

77

99

Answer: E
Concepts:recurrencemodular arithmeticconjugates
Difficulty rating: 2260
Small Hint:

Use a+b=6a+b=6 and ab=1ab=1 to obtain a recurrence for RnR_n

Big Hint:

Compute the recurrence modulo 1010 and look for a short period

Solution:

Because a,ba,b are roots of t2−6t+1=0,t^2-6t+1=0, Rn=6Rn−1−Rn−2. R_n=6R_{n-1}-R_{n-2}. Starting with R0=1, R1=3,R_0=1,\ R_1=3, the units digits are 1,3,7,9,7,3,1,…, 1,3,7,9,7,3,1,\ldots, with period 6.6. Since 12345≡3(mod6),12345\equiv3\pmod6, the units digit is the same as that of R3,R_3, namely 9.9.

Thus the correct answer is E.

← Problem 29#29
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