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Problem 160

AMC 10/12, early questions

Number theory Difficulty 3.7 Find the answer 3rd ASU · Soviet Union

Problem:
Find four different three-digit numbers (in base 10) starting with the same digit, such that their sum is divisible by three of the numbers.

Your answer

Official solution

Solution:
Answer: $108$ , $117$ , $135$ , $180$ . Sum $540 = 108 \times 5 = 135 \times 4 = 180 \times 3$ .

Try looking for a number of the form $3n$ , $4n$ , $5n$ , $n$ . We want $12n$ , $15n$ and $20n$ to have the same first digit. If the first digit is $1$ , this requires $n = 9$ . We must now check that the fourth number which must be $60n - 12n - 15n - 20n = 13n$ also has three digits starting with $1$ . It does, so we are home. [In fact, in this case the first digit must be $1$ , since $20n > \frac{3}{2} 12n$ .]

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