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Factor Dates and Times

February 19, 2016

We’re coming up to 4 April 2016, or 4/4/16.  Notice that, on this day, the day of the month multiplied by the number of the month equals the two digit year i.e.

4 x 4 = 16

That was also the case on 8 February 2016, and will be again on 2 August 2016.  How often do this or similar events happen?

Let’s give the name product dates to these and other combinations of month, day, hours, minutes and/or seconds to match a year number.

Month x day = 2 digit year

Each century has 213 of these dates (see this and other detailed counts in the table below).

We might ask questions such as:

  1. Which year(s) in each century have the most such product dates?
  2. Which years have no such product dates?
  3. In any century, what is the shortest gap between such product dates?
  4. In any century, what is the longest gap between such product dates?

1. Which year has the most product dates?

2024 (and 2124, 2224 etc) has 7 such dates: 24 Jan, 12 Feb, 8 Mar, 6 Apr, 4 Jun, 3 Aug and 2 Dec. The years 2012, 2030, 2036, 2048, 2060 and 2072 get honourable mentions, with 6 dates each.

Why? All of these two digit years are numbers with lots of factors, relative to other numbers of similar size.

All of the honourable mentions except 2012 would have more such dates if our calendar had more months in each year, or more days in particular months.

2. Which years have no product dates?

There are 20 such years: 2037, 2041, 2043, 2047, 2053, 2058, 2059, 2061, 2062, 2067, 2071, 2073, 2074, 2079, 2082, 2083, 2086, 2089, 2094 and 2097.

Why? All of these two digit years are prime or have a factor more than the number of days in a month.

Notes:
– 2058 would not be in this list if it was a leap year i.e. 29 Feb 2058 would be a valid date.
– 2062 would not be in this list if 31 Feb was a valid date

3. In any century, what is the shortest gap between product dates

There are only 16 days from 30 Jan 2030 to 15 Feb 2030.

Honourable mentions go to:

28 Jan 2028 to 14 Feb 2028 (17 days)
26 Jan 2026 to 13 Feb 2026 (18 days)
24 Jan 2024 to 12 Feb 2024 (19 days)
27 Feb 2054 to 18 Mar 2054 (19 days)

Note that the last of these dates is there only because February has only 28 days in that year.

4. In any century, what is the longest gap between product dates?

There are 1097 days from 19 Mar 2057 to 20 Mar 2060.

Honourable mentions go to:

09 Sep 2081 to 28 Mar 2084 (931 days)
06 Dec 2072 to 25 Mar 2075 (839 days)
05 Dec 60 to 21 Mar 63 (836 days)

Each of these (and only these) date pairs surround two consecutive calendar years with no product dates (see 2. above).

The average gap between such dates is just over 171 days or about 5 months 20 days.

Month x day x hour = YYYY

If we want to get the four digit year, we need something more than just month and day values, as the maximum product of month and day is only 31×12=372, a year well before our current year numbering system was developed in 525 by Dionysius Exiguus.

A natural addition is to combine the day and month with the hour to get a four digit year. Only 14 and 22 years this century have a total of 80 and 121 days with combinations of month day and hour like this, using a 12 and 24 hour clock respectively. Of these, 18 and 23 of those combinations occur this year (2016), as in the following table. For example, 24 Apr 2016 at 21h gives us 24 x 4 x 21 = 2016.

24 Apr  2016 at 21h
24 Jul  2016 at 12h 28 Sep  2016 at 08h
28 Apr  2016 at 18h 12 Aug  2016 at 21h 08 Dec  2016 at 21h
16 Jun  2016 at 21h 14 Aug  2016 at 18h 12 Dec  2016 at 14h
21 Jun  2016 at 16h 18 Aug  2016 at 14h 14 Dec  2016 at 12h
24 Jun  2016 at 14h 21 Aug  2016 at 12h 21 Dec  2016 at 08h
28 Jun  2016 at 12h 28 Aug  2016 at 09h 24 Dec  2016 at 07h
16 Jul  2016 at 18h 14 Sep  2016 at 16h 28 Dec  2016 at 06h
18 Jul  2016 at 16h 16 Sep  2016 at 14h

There are 18 dates in 2016 using a 12 hour clock, being the nine in green italics above (both AM and PM).

Why is this year so prolific? Well, 2016 =  25 x 32 x 7 i.e. 8 prime factors – way above the average prime factor count of just over 2 for a number its size. So the separate factor count (including 1 and 2016) is 6 x 3 x 2 = 36. No other number between 2000 and 2099 has as many factors.

Month x day x hour x minute = YYYY

We could even combine month, day, hour and minute to get the four digit year. Only 34 years this century have a total of 3808 and 2644 combinations of month day hour and minute like this, using a 24 and 12 hour clock respectively. 1046 and 689 of them occur this year (2016).

In fact, as I’m publishing this on 19 Feb 2016, the next few occasions of this are on Sun 21 Feb, as follows:

21-Jan-2016 02:48 21 x 1 x 2 x 48 = 2016
21-Jan-2016 03:32 21 x 1 x 2 x 48 = 2016
21-Jan-2016 04:24 21 x 1 x 2 x 48 = 2016
21-Jan-2016 06:16 21 x 1 x 2 x 48 = 2016
21-Jan-2016 08:12 21 x 1 x 2 x 48 = 2016
21-Jan-2016 12:08 21 x 1 x 2 x 48 = 2016
21-Jan-2016 16:06 21 x 1 x 2 x 48 = 2016

All but the last one can be used with a 12 hour clock, but suffixed with each of AM and PM in turn.

Month x day x hour x minute x second = YYYY

Lastly, we could go completely crazy and take this to its logical conclusion, combining month, day, hour, minute and second to get the four digit year. Only 35 years this century have a total of 25848 and 15672 combinations like this, using a 24 and 12 hour clock respectively, of which 8164 and 4797 of them occur this year (2016).

Notes:

  1. There is only one new year added here, compared to the previous list – 2021. The two occasions in that year should be easy to find, and are left as an exercise for the reader.
  2. There would have been 8 more entries on 29 Feb 2030, except that 2030 is not a leap year.

Summary data

The table below sets out both year number factors and event counts for 21st century years. There are some explanatory notes at the foot of the table.

Year

Factor count1

Prime
factors2

Event counts

md =yy3

mdh
=yyyy4

mdhm
=yyyy5
mdhms =yyyy6

hms
=yyyy7

12h 24h 12h 24h 12h 24h 12h 24h

2001

8 31231291 1 1 8 8 36 27 4 4
2002 16 2171111131 2 4 6 64 54 288 204 16

16

2003

1 Prime 2
2004 12 22311671 3

2005

4 514011 2
2006 8 21171591 4 4 4 28 20 8

6

2007

6 322231 2

2008

8 232511 4
2009 6 72411 3 6 3 28 14 8

4

2010

16 213151671 4
2011 1 Prime 2

2012

6 225031 6
2013 8 31111611 1

2014

8 21191531 3 4 4 28 20 8 6
2015 8 51131311 3 1 8 8 36 27 4

4

2016

36 253271 4 18 23 1046 689 8164 4797 168 121
2017 1 Prime 1

2018

4 2110091 5
2019 4 316731 1

2020

12 22511011 5
2021 4 431471 3 4 2 4

2

2022

8 21313371 3
2023 6 71172 1 1 4 6 18 18 2

3

2024

16 23111231 7 4 6 86 71 460 318 32 26
2025 15 3452 2 3 2 116 76 704 413 28

20

2026

4 2110131 2
2027 1 Prime 3

2028

18 2231132 4 2 46 48 334 266 24 20
2029 1 Prime 1

2030

16 215171291 6 4 3 72 40 328 173 28 16
2031 4 316771 1

2032

10 241271 3
2033 4 1911071 2

2034

12 21321131 1
2035 8 51111371 2 12 6 52 26 12

6

2036

6 225091 6
2037 8 3171971

2038

4 2110191 1
2039 1 Prime 1

2040

32 233151171 5 4 10 316 244 2240 1469 80 62
2041 4 1311571

2042

4 2110211 4
2043 6 322271

2044

12 2271731 3
2045 4 514091 3

2046

16 2131111311 1 1 42 25 236 127 16 10
2047 4 231891

2048

12 211 6 1 66 50 660 412 12 10
2049 4 316831 1

2050

12 2152411 3 22 11 124 62 16 8
2051 4 712931 1

2052

24 2233191 2 4 7 228 168 1524 970 72 52
2053 1 Prime

2054

8 21131791 4
2055 8 31511371 2

2056

8 232571 4
2057 6 112171 1 2 2 12 9 36 24 4

3

2058

16 213173 1 2 78 57 480 294 28 22
2059 4 291711

2060

12 22511031 6
2061 6 322291

2062

4 2110311

2063

1 Prime 3
2064 20 2431431 2 68 36 500 256 28

16

2065

8 5171591 1 12 6 52 26 12 6
2066 4 2110331 3

2067

8 31131531 4 4 28 20 8 6
2068 12 22111471 1 22 13 124 68 16

10

2069

1 Prime 1
2070 24 213251231 3 4 9 142 124 832 593 40

36

2071

4 1911091
2072 16 2371371 6 34 20 236 130 20

12

2073

4 316911
2074 8 21171611

2075

6 52831 2
2076 12 22311731 1

2077

4 311671 2
2078 4 2110391 2

2079

16 3371111 8 6 130 76 616 338 28 18
2080 24 2551131 4 4 8 256 192 1820 1171 60

48

2081

1 Prime 2
2082 8 21313471

2083

1 Prime
2084 6 225211 5

2085

8 31511391 1
2086 8 21711491

2087

1 Prime 1
2088 24 2332291 3 8 5 226 121 1384 716 56

32

2089

1 Prime
2090 16 2151111191 5 4 5 72 58 328 224 28

22

2091

8 31171411 1 4 4 28 20 8 6
2092 6 225231 1

2093

8 71131231 1 2 8 12 36 36 4 6
2094 8 21313491

2095

4 514191 1
2096 10 241311 4

2097

6 322331
2098 4 2110491 1

2099

1 Prime 2
2100 36 22315271 10 16 590 397 4056 2391 112

85

21st Century event totals 213 80 121 3808 2644 25848 15672 994 724

Notes for the table above:

  1. Factor counts
    These include 1 and the year number itself. For example, 2001 has factors:
    1
    3
    23
    29
    69 = 3 x 23
    87 = 3 x 29
    667 = 23 x 29 and
    2001 = 3 x 23 x 29
    for a total of 8 factors.
  2. Prime factors
    For example, 2016 = 253271 = 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7
  3. md=yy (month x date = 2 digit year)
    For example, 2004 has dates:
    4 Jan 2004 (4 x 1 = 4)
    2 Feb 2004 (2 x 2 = 4)
    1 Apr 2004 (1 x 4 = 4
  4. mdh=yyyy (month x date x hour = 4 digit year)
    For example, 2002 has 4 (12h clock, AM/PM, italicised) and 6 (24 hour clock) date and hour combinations:
    13 Jul 2002 22h (13 x 7 x 22 = 2002)
    22 Jul 2002 13h (22 x 7 x 13 = 2002)
    26 Jul 2002 11h (26 x 7 x 11 = 2002)
    13 Nov 2002 14h (13 x 11 x 14 = 2002)
    14 Nov 2002 13h (14 x 11 x 13 = 2002)
    26 Nov 2002 07h (26 x 11 x 7 = 2002)
    In these product number types, there tend to be more solutions with a 24 hour clock than those with a 12 hour clock. The reverse happens when minutes and then seconds are also used in the product. Why?
  5. mdhm=yyyy (month x date x hour x minute = 4 digit year)
    For example, 2006 has 4  dates, hours and minutes for either 12 hour or 24 hour clock:
    2 Jan 2006 17:59 (2 x 1 x 17 x 59 = 2006)
    17 Jan 2006 02:59 (17 x 1 x 2 x 59 = 2006)
    1 Feb 2006 17:59 (1 x 2 x 17 x 59 = 2006)
    17 Feb 2006 01:59 (17 x 2 x 1 x 59 = 2006)
  6. mdhms=yyyy (month x date x hour x minute x second = 4 digit year)
    For example, 2009 has 28 (12 hour clock) and 14 (24 hour clock) dates, hours, minutes and seconds:
    7 Jul 2009 1:01:41 (7 x 7 x 1 x 1 x 41 = 2009)
    1 Jul 2009 7:01:41 (1 x 7 x 7 x 1 x 41 = 2009)
    7 Jan 2009 7:01:41 (7 x 1 x 7 x 1 x 41 = 2009)
    1 Jul 2009 1:07:41 (1 x 7 x 1 x 7 x 41 = 2009)
    7 Jan 2009 1:07:41 (7 x 1 x 1 x 7 x 41 = 2009)
    1 Jan 2009 7:07:41 (1 x 1 x 7 x 7 x 41 = 2009)
    1 Jan 2009 1:41:49 (1 x 1 x 1 x 41 x 49 = 2009)
    1 Jul 2009 1:41:07 (1 x 7 x 1 x 41 x 7 = 2009)
    7 Jan 2009 1:41:07 (7 x 1 x 1 x 41 x 7 = 2009)
    7 Jul 2009 1:41:01 (7 x 7 x 1 x 41 x 1 = 2009)
    1 Jan 2009 7:41:07 (1 x 1 x 7 x 41 x 7 = 2009)
    1 Jul 2009 7:41:01 (1 x 7 x 7 x 41 x 1 = 2009)
    7 Jan 2009 7:41:01 (7 x 1 x 7 x 41 x 1 = 2009)
    1 Jan 2009 1:49:41 (1 x 1 x 1 x 49 x 41 = 2009)
    That is, all the above can be used with a 12 hour clock.
  7. hms=yyyy (hour x minute x second = 4 digit year)
    For example, every day(!) in 2006 had 8 (12 hour clock) and 6 (24 hour clock) times:
    01:34:59 (1 x 34 x 59 = 2006)
    01:59:34 (1 x 59 x 34 = 2006)
    02:17:59 (2 x 17 x 59 = 2006)
    02:59:17 (2 x 59 x 17 = 2006)
    17:02:59 (17 x 2 x 59 = 2006)
    17:59:02 (17 x 59 x 2 = 2006)

Teachers may use these sort of dates and times as an extension activity for students, to get them thinking about factors – prime or otherwise

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