"Yes, Bleaug is absolutely correct. Thank you for pointing that out.

If we have

11xmod12 (minute hand) 719xmod12 (second hand)

This should give us the distances from the hour hand, where the hour hand is just xmod12. So if we want a clock with perfect 60 degree angles between each clock, then these set of equations should be satisfied.

11x = 4mod12 719x = 8mod12

x can't be an irrational number, for then the product would be irrational and not 4mod12 or 8mod12. However, we also see that since 11 and 719 are prime, the only rational numbers that can work for this expression are n/11 and t/719, respectively. However, 719 and 11 are coprime, therefore x must be an integer.

i.e. Denominator must be 719*11, but say, divide 719 by 719*11, and you will see that x must be a multiple of 11 in order for the product to be an integer, vice-versa for 11 divided by 719*11.

Since x must be an integer, there are no solutions to this system of equations.

The general solution for the first is 8 + 12n, and the general solution for the second is 4 + 12k."

In that argument, I just had the minute hand 4 units away from the hour hand, and the second hand 8 units away. (12 units in total) The only other possible situation we can have is if the second hand is 4 units away from the hour hand, and the minute hand is 8 units away. But the results shouldn't change.

Here was my answer from the other fox.

ReplyDelete"Yes, Bleaug is absolutely correct. Thank you for pointing that out.

If we have

11xmod12 (minute hand)

719xmod12 (second hand)

This should give us the distances from the hour hand, where the hour hand is just xmod12. So if we want a clock with perfect 60 degree angles between each clock, then these set of equations should be satisfied.

11x = 4mod12

719x = 8mod12

x can't be an irrational number, for then the product would be irrational and not 4mod12 or 8mod12. However, we also see that since 11 and 719 are prime, the only rational numbers that can work for this expression are n/11 and t/719, respectively. However, 719 and 11 are coprime, therefore x must be an integer.

i.e. Denominator must be 719*11, but say, divide 719 by 719*11, and you will see that x must be a multiple of 11 in order for the product to be an integer, vice-versa for 11 divided by 719*11.

Since x must be an integer, there are no solutions to this system of equations.

The general solution for the first is 8 + 12n, and the general solution for the second is

4 + 12k."

In that argument, I just had the minute hand 4 units away from the hour hand, and the second hand 8 units away. (12 units in total) The only other possible situation we can have is if the second hand is 4 units away from the hour hand, and the minute hand is 8 units away. But the results shouldn't change.

http://geometri-problemleri.blogspot.com/2010/11/problem-95-ve-cozumu.html

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