Problem 17 · 2022 AMC 8
Medium
Number Theory
last-digitmod-10
If n is an even positive integer, the double-factorial notation n!! represents the product of all the even integers from 2 to n. For example: 8!! = 2 × 4 × 6 × 8. What is the units digit of the following sum?
2!! + 4!! + 6!! + … + 2022!!
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Answer: B — Units digit 2.
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Hint 1 of 2
“Units digit” means you never need the actual huge values. The moment a product includes 10 as a factor, it ends in 0 — so most of these 1000+ terms are silently zero.
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Hint 2 of 2
Every term from 10!! onward has a factor of 10 (ends in 0), contributing nothing. Only 2!!, 4!!, 6!!, 8!! touch the ones digit.
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Approach: kill the terms ending in 0, keep only the ones digits of the rest
- Insight: the sum runs all the way to 2022!!, but “units digit” lets you ignore almost everything. Any n!! with n ≥ 10 multiplies in a 10, so it ends in 0 and adds nothing to the ones place.
- Only four terms survive: 2!! = 2, 4!! = 8, 6!! = 48, 8!! = 384 — ones digits 2, 8, 8, 4.
- Add just those ones digits: 2 + 8 + 8 + 4 = 22, so the final units digit is 2.
- You'll see this again: for a units-digit question, work mod 10 the whole way — only ones digits feed into ones digits, and a factor of 10 (or 2 and 5 together) zeroes a term out.
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