2011 AMC 8 Problem 22

Attempt Problem 22 of the 2011 AMC 8 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 2011 AMC 8 solutions, or check the answer key.

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

22.

What is the tens digit of $7^{2011}?$

$0$

$1$

$3$

$4$

$7$

Answer: D

Concepts:modular arithmetic modular exponentiation

Difficulty rating: 1520

Solution:

To find the tens digit, it is enough to track powers of $7$ modulo $100$ . Since $7^4=2401\equiv1\pmod{100}$ , the last two digits repeat every four powers.

Because $2011=4\cdot502+3$ , we have $7^{2011}\equiv7^3=343$ $\equiv43\pmod{100}$ . Thus the tens digit is $4$ .

Thus, D is the correct answer.

← Problem 21#21

Full Exam

Problem 23#23 →

Problem 22 in Other Years

1985 AMC 8 · 1986 AMC 8 · 1987 AMC 8 · 1988 AMC 8 · 1989 AMC 8 · 1990 AMC 8 · 1991 AMC 8 · 1992 AMC 8 · 1993 AMC 8 · 1994 AMC 8 · 1995 AMC 8 · 1996 AMC 8 · 1997 AMC 8 · 1998 AMC 8 · 1999 AMC 8 · 2000 AMC 8 · 2001 AMC 8 · 2002 AMC 8 · 2003 AMC 8 · 2004 AMC 8 · 2005 AMC 8 · 2006 AMC 8 · 2007 AMC 8 · 2008 AMC 8 · 2009 AMC 8 · 2010 AMC 8 · 2012 AMC 8 · 2013 AMC 8 · 2014 AMC 8 · 2015 AMC 8 · 2016 AMC 8 · 2017 AMC 8 · 2018 AMC 8 · 2019 AMC 8 · 2020 AMC 8 · 2022 AMC 8 · 2023 AMC 8 · 2024 AMC 8 · 2025 AMC 8 · 2026 AMC 8