Skip to main content

2014 AMC 12B Problem 10

Problem 10 of 25EasierNumber TheoryArithmetic

Danica drove her new car on a trip for a whole number of hours, averaging 5555 miles per hour. At the beginning of the trip, abcabc miles was displayed on the odometer, where abcabc is a 33-digit number with a≥1a \ge 1 and a+b+c≤7.a+b+c \le 7. At the end of the trip, the odometer showed cbacba miles. What is a2+b2+c2?a^2 + b^2 + c^2?

Answer choices

Show solution

Solution

The distance driven is cba−abc=99(c−a),cba - abc = 99(c-a), a multiple of 9.9. Driving a whole number of hours at 5555 mph makes it a multiple of 5555 too, hence a multiple of 495.495. Since the odometer difference is at most a 33-digit number and a≥1,a \ge 1, the distance must be 495,495, so c−a=5.c - a = 5. With a≥1a \ge 1 and a+b+c≤7,a+b+c \le 7, the only choice is a=1,a=1, c=6,c=6, b=0.b=0. Then a2+b2+c2=1+0+36=37.a^2+b^2+c^2 = 1 + 0 + 36 = 37. Thus, the correct answer is D.
AoPS wiki

Tagged: place value · divisibility · least common multiple

More practice