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# Is this number divisible?

## Question

A teacher wrote a large number on the board and asked the students to tell about the divisors of the number one by one.

The 1st student said, "The number is divisible by 2."
The 2nd student said, "The number is divisible by 3."
The 3rd student said, "The number is divisible by 4."
.
.
.
(and so on)
The 30th student said, "The number is divisible by 31.

The teacher then commented that exactly two students, who spoke consecutively, spoke wrongly.

Which two students spoke wrongly?

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16 and 17?

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if the number is 65520, two consecutive numbers are 22 & 23
two students spoke wrongly are 21th and 22th student

if the number is 166320, two consecutive numbers are 25 & 26
two students spoke wrongly are 24th and 25th student

Edited by jasen
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kbrdsk is correct:

The number is 23 * 33 * 52 * 7 * 11 * 13 * 19 * 23 * 29 * 31 = 2,123,581,660,200

This number is not divisible by 16 (24) or 17, but is divisible by all other positive integers less than 32.

Thus, the two students who spoke wrongly were the 15th and 16th students in turn.

jasen's reply has error because more than two students would have spoken wrongly in each case: (Given: Exactly two students spoke wrongly.)

65520 = 24 * 32 * 5 * 7 * 13

Not divisible by 11, 17, 19, 22, 23, 25, 27, 29 or 31

166320 = 24 * 33 * 5 * 7 * 11

Not divisible by 13, 17, 19, 23, 25, 26, 29 or 31

Edited by Logophobic
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Ok I was wrong, but I know the number is 2x3x2x5x7x2x3x11x13x19x23x5x3x29 =  68502634200

Edited by jasen

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