1993 AIME Problems/Problem 6

Problem

What is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?

Solution 1

Denote the first of each of the series of consecutive integers as $a,\ b,\ c$ . Therefore, $n = a + (a + 1) \ldots (a + 8) = 9a + 36 = 10b + 45 = 11c + 55$ . Simplifying, $9a = 10b + 9 = 11c + 19$ . The relationship between $a,\ b$ suggests that $b$ is divisible by $9$ . Also, $10b -10 = 10(b-1) = 11c$ , so $b-1$ is divisible by $11$ . We find that the least possible value of $b = 45$ , so the answer is $10(45) + 45 = 495$ .

Solution 2

Let the desired integer be $n$ . From the information given, it can be determined that, for positive integers $a, \ b, \ c$ :

$n = 9a + 36 = 10b + 45 = 11c + 55$

This can be rewritten as the following congruences:

$n \equiv 0 \pmod{9}$ and $n \geq 36$

$n \equiv 5 \pmod{10}$ and $n \geq 45$

$n \equiv 0 \pmod{11}$ and $n \geq 55$

Since 9 and 11 are relatively prime, n is a multiple of 99. It can then easily be determined that the smallest multiple of 99 with a units digit 5 (this can be interpretted from the 2nd congruence) is 495. This also satisfies the size restrictions, and is thus the answer. $\boxed{495}$

1993 AIME (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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1993 AIME Problems/Problem 6

Contents

Problem

Solution 1

Solution 2

See also