Problem 4 · 2002 AMC 8
Easy
Number Theory
place-value
The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome?
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Answer: B — 4.
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Hint 1 of 2
To stay as close to 2002 as you can, leave the thousands digit at 2 — bumping it would jump you all the way past 3000. So the year looks like 2 ? ? 2.
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Hint 2 of 2
A palindrome is locked by its *front half*: the mirror forces the two middle digits to match. Pick the smallest matching pair that still lands you above 2002.
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Approach: build the next palindrome from the outside in
- Key idea: you only get to *choose* the first half of a palindrome — the back half is its mirror. Keep the leading 2 (raising it overshoots wildly), so the year is 2 ? ? 2.
- The middle two digits must be equal. "00" gives 2002 (not *after*), so the next-smallest equal pair is "11" → 2112. Its digit product is 2 × 1 × 1 × 2 = 4.
- *This transfers:* to find the next palindrome, only nudge the first half upward and mirror it — you never have to scan years one by one.
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