Smallest positive Integer | AIME I, 1993 | Question 6

Find the smallest positive integer that can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers.

Let us take the first of each of the series of consecutive integers as a,b,c

then n=a+(a+1)+...+(a+8)=9a+36=10b+45=11c+55

or, 9a=10b+9=11c+19

or, b is divisible by 9

10b-10=10(b-1)=11c

or, b-1 is divisible by 11

or, least integer b=45

or, 10b+45=10(45)+(45)=495.

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