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The following describes an 8-digit positive integer. Identify this number, and explain the title of this puzzle.

  • The number is in the form of 2021____.
  • It has 24 divisors in total.
  • When its divisors are listed in increasing order, 3 consecutive numbers form an arithmetic progression in six different places (some of them may overlap); two more 3-number arithmetic progressions exist but not consecutively.
  • Its second-largest prime factor, when increased by 1, is a perfect power.

While the use of computer is allowed, an answer with minimal use of computer is preferred.

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  • $\begingroup$ I see that an answer has been provided, but based on intuition about the significance of the number. I'm curious as to whether the solution can be deduced with only a few calculations by hand. (I get the impression you designed the clues so that this could happen.) If this is the case, I'd appreciate it if you could post the 'canonical' solution sometime in future. $\endgroup$
    – COTO
    Commented Dec 11, 2021 at 15:17

1 Answer 1

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The number is in the form of 2021____.

20211225

It has 24 divisors in total.

1, 3, 5, 15, 25, 31, 75, 93, 155, 465, 775, 2325, 8693, 26079, 43465, 130395, 217325, 269483, 651975, 808449, 1347415, 4042245, 6737075, 20211225

I used this website to find the divisors and prime factors https://www.calculatorsoup.com/calculators/math/prime-factors.php
I found everything else manually.

When its divisors are listed in increasing order, 3 consecutive numbers form an arithmetic progression in six different places (some of them may overlap);

1, 3, 5
5, 15, 25
155, 465, 775
8693, 26079, 43465
43465, 130395, 217325
1347415, 4042245, 6737075

two more 3-number arithmetic progressions exist but not consecutively.

31, 93, 155
269483, 808449, 1347415

Its second-largest prime factor, when increased by 1, is a perfect power.

Its prime factors are:
3 x 5 x 5 x 31 x 8693
and 31 + 1 = 32, the 5th power of 2

Everyone is talking about

Christmas! 2021 Dec 25

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  • 1
    $\begingroup$ Care to share how you found the answer? $\endgroup$ Commented Dec 11, 2021 at 7:28
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    $\begingroup$ @StefanLafon Lucky first guess based on the title and the first four digits. Then I saw the seasonal tag and was feeling even surer. Then it was just a matter of checking off the checklist and it checked :) $\endgroup$
    – caPNCApn
    Commented Dec 11, 2021 at 8:18
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    $\begingroup$ It was basically guess and check. I'll add that to the answer. $\endgroup$
    – caPNCApn
    Commented Dec 11, 2021 at 8:22

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