Student divisibility lies

The teacher wrote a number less than 50,000 on the board.
The first student said: This number is divisible by 2
The second student said: This number is divisible by 3
And so on, down to the last one who claimed it was divisible by 13. Two in a row lied.
What number did the teacher write, and which two lied?

Final Answer:

n =  25740

Step-by-step explanation:

$$\begin{gathered} 4=2^2 \\ 5...\text{ prime number} \\ 6=2\cdot 3 \\ 7...\text{ prime number} \\ 8=2^3 \\ 9=3^2 \\ 10=2\cdot 5 \\ 11...\text{ prime number} \\ 12=2^2\cdot 3 \\ 13...\text{ prime number} \\ LCM(4,5,6,7,8,9,10,11,12,13)=2^3\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13=360360 \\ {a}_1=LCM(4,5,6,7,8,9,10,11,12,13)=360360 \\ 2...\text{ prime number} \\ LCM(2,5,6,7,8,9,10,11,12,13)=2^3\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13=360360 \\ {a}_2=LCM(2,5,6,7,8,9,10,11,12,13)=360360 \\ 3...\text{ prime number} \\ LCM(2,3,6,7,8,9,10,11,12,13)=2^3\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13=360360 \\ {a}_3=LCM(2,3,6,7,8,9,10,11,12,13)=360360 \\ LCM(2,3,4,7,8,9,10,11,12,13)=2^3\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13=360360 \\ {a}_4=LCM(2,3,4,7,8,9,10,11,12,13)=360360 \\ LCM(2,3,4,5,8,9,10,11,12,13)=2^3\cdot 3^2\cdot 5\cdot 11\cdot 13=51480 \\ {a}_5=LCM(2,3,4,5,8,9,10,11,12,13)=25740 \\ LCM(2,3,4,5,6,9,10,11,12,13)=2^2\cdot 3^2\cdot 5\cdot 11\cdot 13=25740 \\ {a}_5=LCM(2,3,4,5,6,9,10,11,12,13)=25740 \\ LCM(2,3,4,5,6,7,10,11,12,13)=2^2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13=60060 \\ {a}_6=LCM(2,3,4,5,6,7,10,11,12,13)=60060 \\ LCM(2,3,4,5,6,7,8,11,12,13)=2^3\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13=120120 \\ {a}_7=LCM(2,3,4,5,6,7,8,11,12,13)=120120 \\ LCM(2,3,4,5,6,7,8,9,12,13)=2^3\cdot 3^2\cdot 5\cdot 7\cdot 13=32760 \\ {a}_8=LCM(2,3,4,5,6,7,8,9,12,13)=32760 \\ LCM(2,3,4,5,6,7,8,9,10,13)=2^3\cdot 3^2\cdot 5\cdot 7\cdot 13=32760 \\ {a}_9=LCM(2,3,4,5,6,7,8,9,10,13)=32760 \\ LCM(2,3,4,5,6,7,8,9,10,11)=2^3\cdot 3^2\cdot 5\cdot 7\cdot 11=27720 \\ {a}_{10}=LCM(2,3,4,5,6,7,8,9,10,11)=27720 \\ n=a_5=25740 \end{gathered}$$

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