AMC 10 · 2018 · #19

Grade 7 probability
Archetype Units-Digit Tracking └ Sub-archetype Power Cycle Lookup
units-digit-trackingmodular-arithmetic-mod-10pattern-recognitionprobability-basic pattern-recognitioncasework ↑ Prerequisites: units-digit-trackingprobability-basic
📏 Medium solution 💡 3 insights
Problem
Pick a number m at random from {11,13,15,17,19} and a number n at random from the 20 integers {1999,2000,…,2018}. Find the probability that mⁿ ends in the digit 1.

Pick an answer.

(A)
$\frac{1}{5}$
(B)
$\frac{1}{4}$
(C)
$\frac{3}{10}$
(D)
$\frac{7}{20}$
(E)
$\frac{2}{5}$

AMC 10 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

Tool #5 (Look for a Pattern): the units digit of a power repeats in a short cycle, so instead of computing giant powers we just track that repeating cycle for each base's last digit. Tool #9 (Solve an Easier Related Problem): the giant numbers 11¹⁹⁹⁹ etc. shrink to a question about single last digits. Tool #7 (Identify Subproblems): split the problem into five separate cases, one for each value of m, and find the chance of ending in 1 in each. Tool #2 (Make a Systematic List): list the cycle for each last digit and count how many of the 20 exponents land on a 1.

1STEP 1

Only last digits matter

The units digit of mⁿ depends only on m's last digit, so replace the bases with their last digits 1,3,5,7,9 and study each.

11,13,15,17,19 → last digits 1,3,5,7,9
2STEP 2

Find each cycle

Cycle each last digit: 1 stays 1; 3 and 7 hit 1 only when the exponent is a multiple of 4; 9 hits 1 on even exponents; 5 never hits 1.

3ⁿ:3,9,7,1 | 7ⁿ:7,9,3,1 | 9ⁿ:9,1
3STEP 3

Count winning exponents in 1999..2018

In 1999–2018 there are 5 multiples of 4 and 10 evens, so last digit 1 wins 20, 3 wins 5, 5 wins 0, 7 wins 5, and 9 wins 10.

#{mult. of 4}=5, #{even}=10
4STEP 4

Add up the favorable pairs

The 100 pairs are equally likely; winning counts sum to 20+5+0+5+10=40, so the probability is 40/100 = 2/5, choice (E).

(20+5+0+5+10)/100=40/100=2/5 → (E)
Answer
2/5
The five bases contribute fractions 20/20,5/20,0/20,5/20,10/20, i.e. 1,1/4,0,1/4,1/2. Averaging these five equally likely cases gives 1/5(1+1/4+0+1/4+1/2)=1/5· 2=2/5, matching 40/100. The value 2/5=0.4 sits sensibly between 0 and 1, and it should beat the naive 1/5 guess because the base ending in 1 always wins.
💡Key takeaway

Last digits of powers run in short repeating loops, so count how many exponents land on a 1 for each base, add them up, and divide by all 100 pairs to get 2/5, choice (E).

  • Only last digits matter
  • Find each cycle
  • Count winning exponents in 1999..2018
  • Add up the favorable pairs

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