AMC 10 · 2025 · #9

Grade 8 number-theory

Archetype Modular Cycle Frequency Count └ Sub-archetype Power-Cycle Residue

tens-digit-trackingmodular-arithmeticexponents pattern-recognitioneasier-related-problem ↑ Prerequisites: modular-arithmetic

📏 Medium solution 💡 3 insights

Problem

6⁶⁶ is a power tower: the exponent is itself 6⁶. The finished number has tens of thousands of digits, so it can never be written out. Find the digit that sits in its tens place.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Look for a Pattern

The number itself is out of reach, so I stop chasing it. Only the last two digits decide the tens digit, and the last two digits of a product depend only on the last two digits of the factors — that is a much easier related problem, and it stays small forever. Multiplying by 6 over and over with only two digits kept, the list has to repeat, so I look for the cycle and then figure out where the exponent 46656 lands in it. I split the work into three pieces: evaluate the tower's exponent, find the cycle, locate the exponent inside the cycle.

1STEP 1

Evaluate the exponent first

The exponent is 46656.

6⁶ = 46656 ⟹ 6⁶⁶ = 6⁴⁶⁶⁵⁶

2STEP 2

Trade the number for its last two digits

Track only the last two digits.

N = 100k + r, 0 ≤ r ＜ 100

3STEP 3

Last two digits feed only on last two digits

The last two digits feed only on the last two digits.

6ⁿ⁺¹ = 6ⁿ · 6 (keep only the last two digits each time)

4STEP 4

List the last two digits, find the cycle

Listing them shows a cycle of five.

6¹ → 06, 6² → 36, 6³ → 16, 6⁴ → 96, 6⁵ → 76, 6⁶ → 56, 6⁷ → 36, …

5STEP 5

Locate 46656 in the cycle

46656 sits at the same slot as six to the sixth.

46656 = 5 · 9331 + 1, 6 = 5 · 1 + 1

6STEP 6

Confirm with exponent rules, read off the digit

The last two digits are 56, so the tens digit is 5.

6⁴⁶⁶⁵⁶ = 6⁶ · (6⁵)⁹³³⁰ → 56 × 76 = 4256 → … 56

Answer

The five cycle values 36, 16, 96, 76, 56 have tens digits 3, 1, 9, 7, 5 — exactly the five answer choices, so the entire problem is landing on the correct slot, and no choice can be dismissed on shape alone. The direct check is cheap: 6⁶ = 46656 ends in 56 by ordinary multiplication, and 46656 and 6 leave the same remainder 1 on division by 5, so 6⁴⁶⁶⁵⁶ ends in 56 as well. Tens digit 5, which is choice (C). Guarding against an off-by-one in the cycle length: 6⁷ = 279936 ends in 36, which is where the block restarts, so the period really is 5 and not 4 or 6.

💡Key takeaway

You never need the giant number itself — keep only the last two digits while you multiply, and after five steps they start repeating.

Evaluate the exponent first
Trade the number for its last two digits
Last two digits feed only on last two digits
List the last two digits, find the cycle
Locate 46656 in the cycle
Confirm with exponent rules, read off the digit