Factor Dates and Times
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:
 Which year(s) in each century have the most such product dates?
 Which years have no such product dates?
 In any century, what is the shortest gap between such product dates?
 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 = 2^{5} x 3^{2} 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:
21Jan2016 02:48  21 x 1 x 2 x 48 = 2016 
21Jan2016 03:32  21 x 1 x 2 x 48 = 2016 
21Jan2016 04:24  21 x 1 x 2 x 48 = 2016 
21Jan2016 06:16  21 x 1 x 2 x 48 = 2016 
21Jan2016 08:12  21 x 1 x 2 x 48 = 2016 
21Jan2016 12:08  21 x 1 x 2 x 48 = 2016 
21Jan2016 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:
 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.
 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 21^{st} century years. There are some explanatory notes at the foot of the table.
Year 
Factor count^{1} 
Prime 
Event counts 

md =yy^{3} 
mdh 
mdhm =yyyy^{5} 
mdhms =yyyy^{6} 
hms 

12h  24h  12h  24h  12h  24h  12h  24h  
2001 
8  3^{1}23^{1}29^{1}  1  1  8  8  36  27  4  4  
2002  16  2^{1}7^{1}11^{1}13^{1}  2  4  6  64  54  288  204  16 
16 
2003 
1  Prime  2  
2004  12  2^{2}3^{1}167^{1}  3  
2005 
4  5^{1}401^{1}  2  
2006  8  2^{1}17^{1}59^{1}  4  4  4  28  20  8 
6 

2007 
6  3^{2}223^{1}  2  
2008 
8  2^{3}251^{1}  4  
2009  6  7^{2}41^{1}  3  6  3  28  14  8 
4 

2010 
16  2^{1}3^{1}5^{1}67^{1}  4  
2011  1  Prime  2  
2012 
6  2^{2}503^{1}  6  
2013  8  3^{1}11^{1}61^{1}  1  
2014 
8  2^{1}19^{1}53^{1}  3  4  4  28  20  8  6  
2015  8  5^{1}13^{1}31^{1}  3  1  8  8  36  27  4 
4 

2016 
36  2^{5}3^{2}7^{1}  4  18  23  1046  689  8164  4797  168  121 
2017  1  Prime  1  
2018 
4  2^{1}1009^{1}  5  
2019  4  3^{1}673^{1}  1  
2020 
12  2^{2}5^{1}101^{1}  5  
2021  4  43^{1}47^{1}  3  4  2  4 
2 

2022 
8  2^{1}3^{1}337^{1}  3  
2023  6  7^{1}17^{2}  1  1  4  6  18  18  2 
3 

2024 
16  2^{3}11^{1}23^{1}  7  4  6  86  71  460  318  32  26 
2025  15  3^{4}5^{2}  2  3  2  116  76  704  413  28 
20 
2026 
4  2^{1}1013^{1}  2  
2027  1  Prime  3  
2028 
18  2^{2}3^{1}13^{2}  4  2  46  48  334  266  24  20  
2029  1  Prime  1  
2030 
16  2^{1}5^{1}7^{1}29^{1}  6  4  3  72  40  328  173  28  16 
2031  4  3^{1}677^{1}  1  
2032 
10  2^{4}127^{1}  3  
2033  4  19^{1}107^{1}  2  
2034 
12  2^{1}3^{2}113^{1}  1  
2035  8  5^{1}11^{1}37^{1}  2  12  6  52  26  12 
6 

2036 
6  2^{2}509^{1}  6  
2037  8  3^{1}7^{1}97^{1}  
2038 
4  2^{1}1019^{1}  1  
2039  1  Prime  1  
2040 
32  2^{3}3^{1}5^{1}17^{1}  5  4  10  316  244  2240  1469  80  62 
2041  4  13^{1}157^{1}  
2042 
4  2^{1}1021^{1}  4  
2043  6  3^{2}227^{1}  
2044 
12  2^{2}7^{1}73^{1}  3  
2045  4  5^{1}409^{1}  3  
2046 
16  2^{1}3^{1}11^{1}31^{1}  1  1  42  25  236  127  16  10  
2047  4  23^{1}89^{1}  
2048 
12  2^{11}  6  1  66  50  660  412  12  10  
2049  4  3^{1}683^{1}  1  
2050 
12  2^{1}5^{2}41^{1}  3  22  11  124  62  16  8  
2051  4  7^{1}293^{1}  1  
2052 
24  2^{2}3^{3}19^{1}  2  4  7  228  168  1524  970  72  52 
2053  1  Prime  
2054 
8  2^{1}13^{1}79^{1}  4  
2055  8  3^{1}5^{1}137^{1}  2  
2056 
8  2^{3}257^{1}  4  
2057  6  11^{2}17^{1}  1  2  2  12  9  36  24  4 
3 
2058 
16  2^{1}3^{1}7^{3}  1  2  78  57  480  294  28  22  
2059  4  29^{1}71^{1}  
2060 
12  2^{2}5^{1}103^{1}  6  
2061  6  3^{2}229^{1}  
2062 
4  2^{1}1031^{1}  
2063 
1  Prime  3  
2064  20  2^{4}3^{1}43^{1}  2  68  36  500  256  28 
16 

2065 
8  5^{1}7^{1}59^{1}  1  12  6  52  26  12  6  
2066  4  2^{1}1033^{1}  3  
2067 
8  3^{1}13^{1}53^{1}  4  4  28  20  8  6  
2068  12  2^{2}11^{1}47^{1}  1  22  13  124  68  16 
10 

2069 
1  Prime  1  
2070  24  2^{1}3^{2}5^{1}23^{1}  3  4  9  142  124  832  593  40 
36 
2071 
4  19^{1}109^{1}  
2072  16  2^{3}7^{1}37^{1}  6  34  20  236  130  20 
12 

2073 
4  3^{1}691^{1}  
2074  8  2^{1}17^{1}61^{1}  
2075 
6  5^{2}83^{1}  2  
2076  12  2^{2}3^{1}173^{1}  1  
2077 
4  31^{1}67^{1}  2  
2078  4  2^{1}1039^{1}  2  
2079 
16  3^{3}7^{1}11^{1}  8  6  130  76  616  338  28  18  
2080  24  2^{5}5^{1}13^{1}  4  4  8  256  192  1820  1171  60 
48 
2081 
1  Prime  2  
2082  8  2^{1}3^{1}347^{1}  
2083 
1  Prime  
2084  6  2^{2}521^{1}  5  
2085 
8  3^{1}5^{1}139^{1}  1  
2086  8  2^{1}7^{1}149^{1}  
2087 
1  Prime  1  
2088  24  2^{3}3^{2}29^{1}  3  8  5  226  121  1384  716  56 
32 
2089 
1  Prime  
2090  16  2^{1}5^{1}11^{1}19^{1}  5  4  5  72  58  328  224  28 
22 
2091 
8  3^{1}17^{1}41^{1}  1  4  4  28  20  8  6  
2092  6  2^{2}523^{1}  1  
2093 
8  7^{1}13^{1}23^{1}  1  2  8  12  36  36  4  6  
2094  8  2^{1}3^{1}349^{1}  
2095 
4  5^{1}419^{1}  1  
2096  10  2^{4}131^{1}  4  
2097 
6  3^{2}233^{1}  
2098  4  2^{1}1049^{1}  1  
2099 
1  Prime  2  
2100  36  2^{2}3^{1}5^{2}7^{1}  10  16  590  397  4056  2391  112 
85 

21^{st} Century event totals  213  80  121  3808  2644  25848  15672  994  724 
Notes for the table above:
 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.  Prime factors
For example, 2016 = 2^{5}3^{2}7^{1} = 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7  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  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?  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)  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.  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