Problem 1 · 2024 AMC 8
Easy
Number Theory
last-digitmod-10
What is the ones digit of
222,222 − 22,222 − 2,222 − 222 − 22 − 2 ?
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Answer: B — The ones digit is 2.
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Hint 1 of 2
The question wants only the LAST digit. So why compute all six? What's the smallest piece of each number you actually need?
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Hint 2 of 2
Technique: for any "ones digit of…" question, work only in the ones column. Here every number ends in 2 — that's all that matters.
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Approach: only the ones digit matters
- The question asks for ONE digit, so do ONE digit of arithmetic. Every number ends in 2, and there are five being subtracted, so their ones digits remove 5 × 2 = 10 — an amount ending in 0.
- Taking away something ending in 0 never disturbs a ones digit: 222,222 keeps its 2. This transfers: for any "last digit of…" question, throw away every higher place and work only in the ones column.
- Sanity check: the true value is 222,222 − 24,690 = 197,532 — ones digit 2, as predicted.
Another way — keep the intermediate positive (MAA):
- Look only at the last two digits so the running total never goes negative: 22 − 2 − 2 − 2 − 2 − 2.
- = 22 − 10 = 12. Ones digit: 2.
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