Hour & Minute Coprime: Frequency & Examples

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Discussion Overview

The discussion revolves around the frequency of coprime hour and minute combinations on both 12-hour and 24-hour clocks. Participants explore different interpretations of the question, including the implications of analog versus digital time, and the exclusion of certain values.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that there are many coprime combinations within specific hour ranges, such as 30 times in the hour of 2:00-2:59.
  • There is uncertainty regarding whether the question pertains to a 12-hour or 24-hour clock.
  • One participant proposes that both analog and digital time should be considered, questioning how they differ in terms of coprimality.
  • Another participant argues that all minutes from 00:00-00:59 are coprime, excluding 00:01, but later corrects this to state that only 00:01 is coprime.
  • Participants provide calculations for the number of coprime times per day for both clock formats, with differing results: 865/1440 for the 24-hour clock and 894/1440 for the 12-hour clock.
  • There is a suggestion to move the thread to a more appropriate forum, such as "Number Theory," due to the nature of the question.
  • One participant raises the idea of considering the coprimality of hours, minutes, and seconds together.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of coprime times, the relevance of clock format, and the inclusion of specific values. The discussion remains unresolved with multiple competing perspectives on the topic.

Contextual Notes

Limitations include the lack of consensus on the definitions of coprime in the context of time, the exclusion of certain values, and the implications of analog versus digital representations of time.

BenVitale
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How many times a day are the hour and minute coprime?
 
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Answer
22
 
Caracrist said:
Answer
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
 
Caracrist said:
Answer
22

Sorry. This is wrong.

davee123 said:
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?
 
BenVitale said:
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.

BenVitale said:
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.

BenVitale said:
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 said:
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
 

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