2015 AMC 12B Problem 3

Problem 3 of 25EasierNumber Theory

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is

100,

and one of the numbers is

28.

What is the other number?

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Solution

Write

2x + 3y = 100.

28

were written twice, then

3y = 100 - 56 = 44,

which is not a multiple of

3.

28

is written three times:

2x = 100 - 84 = 16,

giving

x = 8.

Thus, the correct answer is A.

Concepts: Diophantine Equation · divisibility

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.