Problem 6 · 1994 AJHSME
Medium
Number Theory
divisibility
The units digit (one's digit) of the product of any six consecutive positive whole numbers is
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Answer: A — 0.
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Hint 1 of 2
'Any six consecutive numbers' means the answer must work for EVERY such run — so look for something that's always there, no matter where you start.
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Hint 2 of 2
A units digit of 0 just means 'divisible by 10,' and 10 = 2 × 5. So ask: is the product always even, and always a multiple of 5?
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Approach: the product is a multiple of 10
- Six numbers in a row always sweep past a multiple of 5 (you can't go five steps without hitting one) and obviously include even numbers.
- So the product carries both a factor of 5 and a factor of 2 — that's a factor of 10. Anything divisible by 10 ends in 0.
- Why this transfers: 'ends in 0' = 'has both a 2 and a 5 inside.' You'll use this same 2-and-5 logic to count trailing zeros in factorials and big products. Notice we never multiplied anything — spotting the guaranteed factors beats computing the giant product.
Mark:
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