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Saturday, April 17, 2021

2021/022)if S(n) is sum of digits of n find n for which n + S(n) = 2021

 As n+ S(n) = 2021 so n < 2021.

Now highest S(n) for number n < 2021 is 28 that is when 

n = 1999.

so n >= 2021-28 or n>=1993

Now working mod 9 we have n \equiv S(n) \pmod 9

So n + S(n) \equiv 2n \pmod 9

so 2n \equiv 2021 \pmod 9

or 2n \equiv 5 \pmod 9

as 5 is odd add 9 to get even

so 2n \equiv 14 \pmod 9

or n \equiv 7 \pmod 9

so we need to check for candidates between 1993 and 2021 which are 7 mod 9 and they are 1996, 2005,2014 out of which 1996 and 2014 satisfy the condition 

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