Hour & Minute Coprime: Frequency & Examples

[BenVitale]

How many times a day are the hour and minute coprime?

 

[Caracrist]

Answer

Spoiler

22

 

[davee123]

Answer

Spoiler

22

If I understand correctly, it's a lot more than that! There are 30 times within the hour of 2:00-2:59 alone that are co-prime!

Is this question meant on a 24-hour clock or a 12-hour clock?

DaveE

 

[BenVitale]

Answer

Spoiler

22

Sorry. This is wrong.

If I understand correctly, it's a lot more than that! There are 30 times within the hour of 2:00-2:59 alone that are co-prime!

Is this question meant on a 24-hour clock or a 12-hour clock?

DaveE

Right. There are more than that.

You need to consider 2 cases:
- Analog time
- Digital time

And, you need to exclude 0.


Perhaps, I was wrong to post this question in this forum. Could the moderators transfer/repost this thread in the "Number Theory" forum?

 

[davee123]

You need to consider 2 cases:
- Analog time
- Digital time

I'm not sure I understand why Analog and Digital time would be different, unless you mean a 24-hour digital clock (IE, one that says 23:17 rather than 11:17pm). If we're talking a 12-hour clock, then the answer is all the coprime times in the range 1:00-12:59, then doubled. If we're talking a 24-hour clock, then it's just all the coprime times in the range 00:00-23:59.

And, you need to exclude 0.

I'm not sure I understand why? My reference being Wikipedia:

"The numbers 1 and −1 are coprime to every integer, and they are the only integers to be coprime with 0."

Hence, in a 24-hour clock, ALL the minutes from 00:00-00:59 are coprime, excluding 00:01.

Perhaps, I was wrong to post this question in this forum. Could the moderators transfer/repost this thread in the "Number Theory" forum?

I dunno, word problems frequently go here, although I'm not sure what the official stance on them is.

DaveE

 

[davee123]

Hence, in a 24-hour clock, ALL the minutes from 00:00-00:59 are coprime, excluding 00:01.

Whoops, got that backwards-- none of the minutes from 00:00-00:59 are coprime, except for 00:01, but ALL of the minutes from 01:00-01:59 are coprime.

Since there seems to be no resolution, here's both answers:

24-hour clock: 865/1440 times per day are coprime

12-hour clock: 894/1440 times per day are coprime (or 447/720 distinct combinations)

And how about the hour, minute, and seconds all being coprime of each other?

24-hour clock: 24594/86400 times per day are coprime

12-hour clock: 25252/86400 times per day are coprime (or 12626/43200 distinct combinations)

DaveE