Question

James brought fewer donuts than two dozen of donuts. He wants to evenly share them among 2,3,4 people. No matter how many people he shares them with he Aways have one left over. How many did he buy? What numbers goes in to 24 when factored with 2,3,4 gives remainder of 1

Answer 1

Answer on question #35321 – Math – Number Theory

James brought fewer donuts than two dozen of donuts. He wants to evenly share them among 2, 3, 4 people. No matter how many people he shares them with he always have one left over. How many did he buy? What numbers goes in to 24 when factored with 2, 3, 4 gives remainder of 1

Answer

Let $x$ is a number of donuts. We know that $x < 24$.

We have the following system of congruence relations

$$\left\{ \begin{array}{l} x \equiv 1 \pmod{2} \\ x \equiv 1 \pmod{3} \\ x \equiv 1 \pmod{4} \end{array} \right.$$

From the first congruence relation we get

$$x = 2t + 1$$

Substitute this into the second congruence relation

$$2t + 1 \equiv 1 \pmod{3} \Rightarrow 2t \equiv 0 \pmod{3} \Rightarrow t \equiv 0 \pmod{3}$$ $$t = 3t_1 \Rightarrow x = 6t_1 + 1$$

Substitute it into the third equation

$$6t_1 + 1 \equiv 1 \pmod{4} \Rightarrow 6t_1 \equiv 0 \pmod{4} \Rightarrow t_1 \equiv 0 \pmod{2}$$ $$t_1 = 2t_2 \Rightarrow x = 12t_2 + 1,$$

where $t_2$ is integer. Therefore, $x = 13, 25, 37, \ldots$. According to condition $x < 24$ we get that number of donuts is 13.

Answer: 13.