2013 AMC 12B Problem 14

Problem 14 of 25IntermediateAlgebra Number Theory

Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is

N.

What is the smallest possible value of

N?

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Solution

A sequence starting

a_1, a_2

has seventh term

5a_1 + 8a_2.

For the two sequences,

5a_1 + 8a_2 = 5b_1 + 8b_2,

5(b_1 - a_1) = 8(a_2 - b_2).

Since

\gcd(5, 8) = 1,

we need

b_1 - a_1

to be a multiple of

8

and

a_2 - b_2

to be a multiple of

5.

Taking

a_1 \lt b_1

with nondecreasing terms gives

a_1 \le b_1 - 8 \le b_2 - 8 \le a_2 - 13.

Choosing

a_1 = 0,

b_1 = b_2 = 8,

a_2 = 13

yields

N = 5\cdot 0 + 8\cdot 13 = 104.

Thus, the correct answer is C.

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