2014 AMC 10B Problem 14

Problem 14 of 25IntermediateNumber Theory

Danica drove her new car on a trip for a whole number of hours, averaging

55

miles per hour. At the beginning of the trip,

abc

miles was displayed on the odometer, where

abc

is a

3

-digit number with

a\ge1

and

a+b+c\le7.

At the end of the trip, the odometer showed

cba

miles. What is

a^2+b^2+c^2?

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Solution

We know that the difference of the numbers

cba

and

abc

is equal to:

100c + 10b+a - 100a - 10b-c

= 99(c-a)

We know that this number also must be a multiple of

55.

\gcd(55,99)

11,

we know that

c-a

is a multiple of

5,

and

c > a.

This makes

a = 1, b = 0, c = 6

the only possible value with

a+ b+c \leq 7

as every other combination has

a+b+c > 7.

As such,

a^2+b^2+c^2 = 37.

Thus, the correct answer is D .