, 2003

Number theory Difficulty 4.4 AIME Find the answer Italy

Problem:

Determine the number of quadruples of integers (not necessarily distinct) between $1$ and $12$ (inclusive) that satisfy all of the following conditions:
- the sum of the first two numbers is even
- the sum of the first three numbers is a multiple of $3$
- the sum of the four numbers is a multiple of $4$ .
(Two quadruples that differ even only in the order of the addends are to be considered distinct).

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

The answer is $864$ . For each choice of the first number, the second number can be chosen in $6$ ways to satisfy the first condition; again, for each possible sum of the first $2$ numbers there exist $4$ possible choices for the third so that the second condition is satisfied; finally, for each possible choice of the first $3$ numbers there exist $3$ possibilities for the fourth. Hence the total number of possibilities is $12 \cdot 6 \cdot 4 \cdot 3 = 864$ .

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