Divisible 79464

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?

Correct answer:

n = 25740

Step-by-step explanation:

4 = 2^{2} 5 . . . prime number 6 = 2 \cdot 3 7 . . . prime number 8 = 2^{3} 9 = 3^{2} 10 = 2 \cdot 5 11 . . . prime number 12 = 2^{2} \cdot 3 13 . . . prime number L C M (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} = L C M (4, 5, 6, 7, 8, 9, 10, 11, 12, 13) = 360360 2 . . . prime number L C M (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} = L C M (2, 5, 6, 7, 8, 9, 10, 11, 12, 13) = 360360 3 . . . prime number L C M (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} = L C M (2, 3, 6, 7, 8, 9, 10, 11, 12, 13) = 360360 L C M (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} = L C M (2, 3, 4, 7, 8, 9, 10, 11, 12, 13) = 360360 L C M (2, 3, 4, 5, 8, 9, 10, 11, 12, 13) = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13 = 51480 a_{5} = L C M (2, 3, 4, 5, 8, 9, 10, 11, 12, 13) = 25740 L C M (2, 3, 4, 5, 6, 9, 10, 11, 12, 13) = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13 = 25740 a_{5} = L C M (2, 3, 4, 5, 6, 9, 10, 11, 12, 13) = 25740 L C M (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} = L C M (2, 3, 4, 5, 6, 7, 10, 11, 12, 13) = 60060 L C M (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} = L C M (2, 3, 4, 5, 6, 7, 8, 11, 12, 13) = 120120 L C M (2, 3, 4, 5, 6, 7, 8, 9, 12, 13) = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 = 32760 a_{8} = L C M (2, 3, 4, 5, 6, 7, 8, 9, 12, 13) = 32760 L C M (2, 3, 4, 5, 6, 7, 8, 9, 10, 13) = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 = 32760 a_{9} = L C M (2, 3, 4, 5, 6, 7, 8, 9, 10, 13) = 32760 L C M (2, 3, 4, 5, 6, 7, 8, 9, 10, 11) = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 = 27720 a_{10} = L C M (2, 3, 4, 5, 6, 7, 8, 9, 10, 11) = 27720 n = a_{5} = 25740

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