AMC 10 · 2016 · #8

Grade 5 arithmetic
Archetype Modular Cycle Frequency Count └ Sub-archetype Power-Cycle Residue
modular-arithmeticexponentsunits-digit-tracking pattern-recognition ↑ Prerequisites: modular-arithmetic
📏 Medium solution 💡 3 insights
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Problem
Find the tens digit of the number 2015²⁰¹⁶ - 2017.

Pick an answer.

(A)
0
(B)
1
(C)
3
(D)
5
(E)
8

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

We cannot write out 2015²⁰¹⁶ — it has thousands of digits. But the question only asks for the tens digit, and the tens digit depends only on the last two digits of a number. Tool #9 (Solve an Easier Related Problem) lets us throw away everything except the last two digits at every step, turning a monster power into a two-digit chase. Once the problem is that small, Tool #5 (Look for a Pattern) shows that the last two digits of the powers of 15 settle into a short repeating cycle, so we never multiply 2016 times. Tool #7 (Identify Subproblems) splits the job into two clean pieces: first get the last two digits of the power, then subtract 2017.

1STEP 1

Only the last two digits matter

The tens digit sits second from the right, so it is fixed by the last two digits — ignore everything further left.

tens digit of N = tens digit of (last two digits of N)
2STEP 2

Replace 2015 with 15

Products keep only the last two digits of their factors, so 2015²⁰¹⁶ ends like 15²⁰¹⁶ — work with 15, not 2015.

2015²⁰¹⁶ ≡ 15²⁰¹⁶ (mod 100)
3STEP 3

Find the pattern in powers of 15

Listing powers of 15, the last two digits alternate: even powers end in 25, odd powers end in 75.

15² → 25, 15³ → 75, 15⁴ → 25, 15⁵ → 75, …
4STEP 4

Apply the pattern at exponent 2016

The exponent 2016 is even, so 15²⁰¹⁶ — and therefore 2015²⁰¹⁶ — ends in 25.

2016 is even → 15²⁰¹⁶ → 25 → 2015²⁰¹⁶ ≡ 25 (mod 100)
5STEP 5

Subtract 2017 and read the tens digit

Subtracting 2017 disturbs only the tail: 25 - 17 = 08, so the tens digit is 0 and the answer is (A).

… 25 - 2017 → 25 - 17 = 08 → tens digit = 0 = (A)
Answer
0
Sanity-check the cycle with a smaller even exponent: 15⁴ = 50625, whose last two digits are 25, matching the rule for even powers. Subtracting a number ending in 17 from one ending in 25 gives an ending of 08, so a tens digit of 0 is consistent. The result 0 is choice (A).
💡Key takeaway

If a question only wants the last digits, keep just the last two digits at every step and hunt for the pattern in the powers.

  • Only the last two digits matter
  • Replace 2015 with 15
  • Find the pattern in powers of 15
  • Apply the pattern at exponent 2016
  • Subtract 2017 and read the tens digit

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