Problem 30 · 2023 Math Kangaroo
Stretch
Number Theory
factorizationcareful-counting
The product of six consecutive numbers is a 12-digit number of the form abb cdd cdd abb, where the digits a, b, c and d are also consecutive numbers in any order. What is the value of the digit d?
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Answer: C — 3
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Hint 1 of 2
The product of six consecutive numbers is a 12-digit number, which narrows the starting value a lot.
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Hint 2 of 2
Search near that range and match the repeating abb cdd cdd abb digit pattern.
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Approach: bound the starting value, then match the digit pattern
- A 12-digit product of six consecutive integers forces the run to start in the mid-70s.
- Testing those, 74·75·76·77·78·79 = 200 133 133 200, of the form abb cdd cdd abb with a=2, b=0, c=1, d=3.
- The digits 0, 1, 2, 3 are consecutive, and d = 3.
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