Competition problem — Number theory

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Number theory Difficulty 4.5 AIME Find the answer Italy

Problem:

An ancient text states that Methuselah lived 150 years, where the symbol $\ast$ replaces the units digit, which scholars are unable to read. Fortunately we have three other manuscripts about Methuselah's life; the first claims that he lived an even number of years, the second that he lived a number of years that is a multiple of 3, the third that he lived a number of years that is a multiple of 5. Knowing that exactly one of these three manuscripts contains false information, how many different digits could be hidden behind the symbol $\ast$ ?

Pick one

Solution

Solution:

(a) $150\ast$ is a multiple of 2 and 3, but not of 5;
(b) $150\ast$ is a multiple of 2 and 5, but not of 3;
(c) $150\ast$ is a multiple of 3 and 5, but not of 2.

First of all, notice that $1500$ is a multiple of $2 \cdot 3 \cdot 5 = 30$ , and there are no other multiples of $10$ or $15$ between $1500$ and $1509$ . The last two cases are therefore excluded. The only multiple of $6$ that is not also a multiple of $5$ among the $10$ possible values is $1506$ , so only the first of the three cases can be the correct one, and there is only one possible choice for the missing digit.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.