Competition problem — Number theory

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Number theory Difficulty 3.7 AMC 10/12 Find the answer Slovenia

Which digit of the 7-digit number $2345678$ should we delete to get a 6-digit number divisible by $9$ ?

Pick one

Solution

A positive integer is divisible by $9$ if and only if the sum of its digits is divisible by $9$ . The sum of the digits of $2345678$ is equal to $35 = 3 \cdot 9 + 8$ , so we should delete the digit $8$ . We obtain the 6-digit number $234567$ . The sum of its digits is equal to $27 = 3 \cdot 9$ , so it must be divisible by $9$ .

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