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Naoish numbers (Posted on 2020-11-16) Difficulty: 3 of 5
We say an integer n is naoish if n≥90 and the second-to-last digit of n (in decimal notation) is equal to 9. For example, 10798, 1999 and 90 are naoish, whereas 9900, 2009 and 9 are not. Nino expresses 2020 as a sum:

2020=n1+n2+...+nk

where each of the nj is naoish. What is the smallest positive number k for which Nino can do this?

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 2.0000 (1 votes)

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Solution Full solution Comment 1 of 1
k cannot equal 2, because 2*9 = 18, and we would need a carry of 4 from the addition of the 2 units digits, which is impossible.

k cannot equal 3, because 3*9 = 27, and we would need a carry of 5 from the addition of the 2 units digits, which is impossible.

Similarly, k cannot equal 4,5,6 or 7, because we would need a carry of k+2 from the addition of the k units digits, which is impossible

k can equal 8, because 8*9 = 72, and we would need a carry of 0 from the addition of the units digits.

For instance, 90+90+90+90+90+90+90+1390 = 2020

Edited on November 16, 2020, 12:51 pm
  Posted by Steve Herman on 2020-11-16 10:27:07

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