2020 AMC 12A Problem 4

Below is the professionally curated solution for Problem 4 of the 2020 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2020 AMC 12A solutions, or check the answer key.

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Concepts:multiplication principle divisibility digits

Difficulty rating: 1200

How many $4$ -digit positive integers (that is, integers between $1000$ and $9999,$ inclusive) having only even digits are divisible by $5?$

$80$

$100$

$125$

$200$

$500$

Solution:

To be divisible by $5$ the last digit is $0$ or $5,$ and to be even it must be $0.$ So the units digit is fixed.

The leading digit is a nonzero even digit: $2, 4, 6, 8$ give $4$ choices. Each of the two middle digits is any even digit $0, 2, 4, 6, 8,$ giving $5$ choices each.

The total is $4 \cdot 5 \cdot 5 \cdot 1 = 100.$

Thus, B is the correct answer.

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