Problem 12 · 1995 AJHSME
Hard
Number Theory
factor-pairs
A lucky year is one in which at least one date, written as month/day/year, has the property that the month times the day equals the last two digits of the year. For example, 1956 is lucky because 7/8/56 has 7 × 8 = 56. Which of the following is NOT a lucky year?
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Answer: E — 1994.
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Hint 1 of 2
'Month × day = the two-digit year' is really asking: can you FACTOR that number into month × day, where the month is 1–12 and the day is a real calendar day? So this is a factoring hunt in disguise.
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Hint 2 of 2
The dangerous factor pairs are ones where the only factorizations need a month bigger than 12. Check 94 last — it's the one with almost no factors.
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Approach: treat each year as a factoring question: month (1–12) × day
- Reframe 'lucky' as: can the last two digits be written as (a number 1–12) × (a valid day)? That turns the puzzle into checking factor pairs.
- 90 = 9 × 10 ✓, 91 = 7 × 13 ✓, 92 = 4 × 23 ✓, 93 = 3 × 31 ✓ — each has a month ≤ 12 paired with a real day.
- 94 is the trap: its only factorizations are 1 × 94 and 2 × 47. The month must be 1–12, so the only candidate is month 2 (February) with day 47 — impossible. No valid pair exists, so 1994 is not lucky.
- Why this transfers: when a number has few factors (here 94 = 2 × 47, a product of two primes), it has very few ways to be split — which is exactly why it fails. Always factor first; the sparse-factor number is your prime suspect.
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