The tickets

The tickets to the show cost an integer greater than 1. Also, the sum of the price of the children's and adult tickets and their products was the power of the prime number. Find all possible ticket prices.

Correct answer:

a₁ =  2
a₂ =  4
a₃ =  8
a₄ =  16

Step-by-step explanation:

$$\begin{gathered} a+b=p_1^x \\ a\cdot b=p_2^y \\ {p}_1,p_2\dots prime \\ {n}_2=2,4,6,8\dots .,2k \\ a=b=p_2 \\ a+b=2p_2=p_1^x \\ =>p_1=2 \\ {p}_1\cdot p_2=p_1^x \\ {a}_1=2 \\ {b}_1=2 \\ {s}_1=2+2=4=2^2 \\ {s}_2=2\cdot 2=4=2^2 \end{gathered}$$

$$\begin{gathered} a_2=4 \\ {b}_2=4 \\ {s}_1=4+4=8=2^3 \\ {s}_2=4\cdot 4=16=2^4 \end{gathered}$$

$$\begin{gathered} a_3=8 \\ {b}_3=8 \\ {s}_1=8+8=16=2^4 \\ {s}_2=8\cdot 8=64=2^6 \end{gathered}$$

$$\begin{gathered} a_4=16 \\ {b}_4=16 \\ {s}_1=16+16=32=2^5 \\ {s}_2=16\cdot 16=256=2^8 \end{gathered}$$

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