A four-digit number can be made by repeating a two-digit number. For example, 1111 is made by repeating 11, and 1919 is made by repeating 19. How many such numbers are there between 2000 and 10 000?
CEMC Gauss (Grade 7), 2020
Pick one
Solution
Each four-digit number must be greater than 2000, and so the smallest two-digit number that may be repeated is 20 (to give 2020).
Each four-digit number must be less than 10 000, and so the largest two-digit number that may be repeated is 99 (to give 9999).
Each two-digit number between 20 and 99 may be repeated to give a four-digit number between 2000 and 10 000.
In total, there are numbers that satisfy the given conditions.
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