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# The most productive year?

## Question

Earlier this year I noticed that the date could be written as 1.13.13 [January 13, 2013] and

the day and month multiplied gave the year [modulo 100] Thirteen being both prime and

greater then the number of months, this is the only such date in 2013.

What year this century contains the most such dates?

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Year 2024.

Has Seven instances when product of date and month is the year (modulo 100)
1.24.24
2.12.24
3.08.24
4.06.24
6.04.24
8.03.24
12.02.24
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There are two years which have the maximum number of three such dates:

2017 with dates 1/17/17, 7/31/17, 9/13/17 and 2019 with dates 1/19/19, 7/17/19, 11/29/19.

If, however, we drop the prime and larger than 12 requirements for the year modulo 100, the year
2020 contains the most such dates, 2040 comes in close behind. The 13 such dates in 2020 are:
1/20/20, 2/10/20, 4/5/20, 4/30/20, 5/4/20, 5/24/20, 6/20/20, 8/15/20, 10/2/20, 10/12/20,
10/22/20, 11/20/20, 12/10/20.
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What I tried to convey by modulo 100 simply was 2013 becomes 13, which then is the straight product of 1 and 13.

Being prime and all that is not a requirement, just dates like 5/12/60.

You've opened up a richer set of possibilities, which I will now go back and work through.

Thanks.

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