Template:Sequence of the Day for October 31

Intended for: October 31, 2011

Timetable

• First draft entered by Alonso del Arte on June 28, 2011 (as a near verbatim copy of the write-up from October 31, 2010) ✓
• Draft reviewed by Daniel Forgues on October 31, 2012 ✓
• Draft to be approved by September 30, 2011

Yesterday's SOTD * Tomorrow's SOTD

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A008595: Multiples of 13.

{13, 26, 39, 52, 65, 78, 91, ...}

In this day and age, there are still people afraid of the number 13. That’s why I’ve chosen this sequence for Sequence of the Day on Halloween.

A decimal rule for divisibility by 13:

To find out wether a number is a multiple of 13 (and/or a multiple of 7 and/or 11 for that matter), consider

1001 = 1111  −  110 = 11  ×  101  −  11  ×  10 = 11  ×  91 = 7  ×  11  ×  13,

which means that if you take, starting from the right, the decimal digits by groups of three (giving “digits” base 1000) and alternatively add and subtract the obtained three digits numbers, iterating the procedure will lead to a three digit number that is divisible by 13 (and/or by 7 and/or by 11).

For example, with 183440731043453, we have

+453  −  043 + 731  −  440 + 183 = 884 = 68  ×  13 = 2 2  ×  13  ×  17,

so 183440731043453 (= 13 × 23 2 × 37 3 × 61 × 89 × 97) is a multiple of 13, but neither a multiple of 7 nor a multiple of 11.

With 14124936290345881, we have

+ 881  −  345 + 290  −  936 + 124  −  14 = 0,

so 14124936290345881 (= 7 × 11 × 13 × 23 2 × 37 3 × 61 × 89 × 97) is a multiple of 7, 11 and 13.