AMC 10 · 2006 · #11

Grade 5 arithmetic
Archetype Units-Digit Tracking └ Sub-archetype Tail Collapses to Zero
factorialmodular-arithmeticplace-value identify-subproblems ↑ Prerequisites: factorialmodular-arithmetic
📏 Medium solution 💡 3 insights
📘 View easy version →
Problem
Add up every factorial from 7! through 2006!, that is 7!+8!+9!+…+2006!. What is the tens digit of this enormous total?

Pick an answer.

(A)
1
(B)
3
(C)
4
(D)
6
(E)
9

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Solve an Easier Related Problem

The sum stretches to 2006!, a number with thousands of digits, so Tool #9 (Solve an Easier Related Problem) is the key: because only the tens digit is asked, almost every term can be thrown away, shrinking the problem to a tiny sum. Tool #5 (Look for a Pattern) supplies the reason it shrinks — factorials pick up more and more trailing zeros as they grow, so from 10! onward each term ends in 00 and cannot touch the tens digit. Tool #16 (Change Focus) keeps attention on just the last two digits at every step, so no giant multiplications or additions are ever needed.

1STEP 1

See why 10! ends in two zeros

Inside 10! the factors 5 and 10 meet plenty of even factors, so 10! is a multiple of 100: 10!=3628800, ending in 00.

10!=3628800≡ 00 (mod 100)
2STEP 2

Every larger factorial keeps those zeros

Each later factorial is 10! times more whole numbers, and multiplying keeps trailing zeros, so 11!,12!,…,2006! all end in 00.

n!=n×(n-1)×…×11 × 10!≡ 00 (mod 100) (n ≥ 10)
3STEP 3

Keep only the three terms that matter

So only the terms below 10! decide the tens digit: 7!=5040, 8!=40320, 9!=362880.

7!=5040, 8!=40320, 9!=362880
4STEP 4

Add just the last two digits

Their last two digits give 40+20+80=140, so the sum ends in 40 and the tens digit is 4, choice (C).

40+20+80=140≡ 40 (mod 100) → tens digit=4 (C)
Answer
4
Check the discarded terms really are safe: 10!=3628800 ends in 00, and every factorial after it is a multiple of 10!, so it also ends in 00 — confirmed, they cannot affect the tens digit. Check the surviving arithmetic against the full numbers: 5040+40320+362880=408240, which indeed ends in 40 with tens digit 4, matching the shortcut. The answer 4 is choice (C), one of the listed options, so it is consistent.
💡Key takeaway

When only the last digits matter, drop every number that ends in zeros — here all factorials from 10! up vanish, leaving just 7!+8!+9!, which ends in 40, so the tens digit is 4.

  • See why 10! ends in two zeros
  • Every larger factorial keeps those zeros
  • Keep only the three terms that matter
  • Add just the last two digits

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