Problem 27 · 2023 Math Kangaroo
Stretch
Number Theory
Arithmetic & Operations
digit-sumcasework
Bart wrote the number 1015 as a sum of numbers that are made up of only the digit 7. In total he used the digit 7 ten times (see diagram). Now he wants to write the number 2023 as a sum of numbers made up of only the digit 7, using the digit 7 nineteen times in total. How many times does he have to use the number 77?
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Answer: E — 6
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Hint 1 of 2
Numbers made only of 7s are 7, 77, 777, ... each contributing several 7s to the digit-count.
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Hint 2 of 2
Match the value 2023 and the total of 19 sevens together; that pins how many 77s appear.
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Approach: balance the value 2023 and the count of 19 sevens
- Let there be a parts equal to 7, b equal to 77 and c equal to 777 (777 × 3 > 2023, so no bigger parts).
- Value: 7a + 77b + 777c = 2023, i.e. a + 11b + 111c = 289 (dividing by 7). Digit count: a + 2b + 3c = 19.
- Subtracting gives 9b + 108c = 270, so b + 12c = 30; the only choice keeping a ≥ 0 is c = 2, b = 6 (then a = 1).
- Check: 7 + 6×77 + 2×777 = 7 + 462 + 1554 = 2023, using 1 + 12 + 6 = 19 sevens. So 77 is used 6 times, option E.
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