1968 AMC 12 Problem 26

Attempt Problem 26 of the 1968 AMC 12 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 1968 AMC 12 solutions, or check the answer key.

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

26.

Let $S=2+4+6+\cdots+2N,$ where $N$ is the smallest positive integer such that $S\gt1{,}000{,}000.$ Then the sum of the digits of $N$ is:

$27$

$12$

$6$

$2$

$1$

Answer: E

Concepts:arithmetic sequence inequality digits

Difficulty rating: 1510

Small Hint:

The sum is $N(N+1)$

Big Hint:

Compare $999\cdot1000$ and $1000\cdot1001$ with one million

Solution:

We have $S=N(N+1).$ For $N=999,$ this is $999{,}000\lt1{,}000{,}000,$ while for $N=1000$ it is $1{,}001{,}000.$ Thus $N=1000,$ whose digit sum is $1.$

Therefore, the correct answer is E.

← Problem 25#25

Full Exam

Problem 27#27 →

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