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Properties of 153

From "Figuring - The Joy of Numbers" by Shakuntala Devi, pages 125 and 126.

Many sources state that 153 equals 1 3 + 5 3 + 3 3
[ Mike Keith speaks about Wild Narcissistic Numbers ]
or that
153 = 1! + 2! + 3! + 4! + 5! or that
153 is the 17th triangular number (1+2+3+4+...+15+16+17)

but the most magic moment occurs when working out its reciprocal
1  ÷  153 = 0,006535947712418300653594...
"Take all the significant figures, multiply by 17, and multiples of 17, and watch the pattern formed :"

65359477124183 x 17 = 1111111111111111
65359477124183 x 34 = 2222222222222222
65359477124183 x 51 = 3333333333333333
65359477124183 x 68 = 4444444444444444
65359477124183 x 85 = 5555555555555555
65359477124183 x 102 = 6666666666666666
65359477124183 x 119 = 7777777777777777
65359477124183 x 136 = 8888888888888888
65359477124183 x 153 = 9999999999999999

I just love these palindromic side-effects !

Shyam Sunder Gupta maintains a page with lots and lots more
facts and figures about the number 153.
CURIOUS PROPERTIES OF 153

[ May 1, 2003 ]
Samuel Cheng (email) notes that " this property
is nothing special for 153. It can be proved true
for any number, if its reciprocal is a repeating number.
E.g. 1/63 = 0,0158730... and 15873 x 63 = 999999 ! "

Samuel, you're right of course. It seems I was a bit
over-enthusiastic at the time and neglected your generalization.

[ July 15, 2003 ]
Patrick Capelle from Brussels (email) writes that it is a property
of the division by 9 !

My analysis of the observation about the reciprocal of 153 :

153 = 9 x 17. In consequence,

(1/153) x 17 = (1/(9 x 17)) x 17 = 1/9 = 0,11111111111...
(1/153) x 34 = (1/(9 x 17)) x 34 = 2/9 = 0,22222222222222...
(1/153) x 51 = (1/(9 x 17)) x 51 = 3/9 = 0,33333333333333...
(1/153) x 68 = (1/(9 x 17)) x 68 = 4/9 = 0,44444444444444...
(1/153) x 85 = (1/(9 x 17)) x 85 = 5/9 = 0,55555555555555...
(1/153) x 102 =(1/(9 x 17)) x 102 = 6/9 = 0,6666666666666...
(1/153) x 119 = (1/(9 x 17)) x 119 = 7/9 = 0,7777777777777...
(1/153) x 136 = (1/(9 x 17)) x 136 = 8/9 = 0,8888888888888...
(1/153) x 153 = (1/(9 x 17)) x 153 = 9/9 = 1 = 0,9999999999...

All the results are multiples of 1/9 = 0,111111... ,
and are not directly in relation with the presence of 153.
You have taken a part of the period of 1/153, instead of 1/153,
and you have also changed the units.
In conclusion, what you give is not a property of the reciprocal
of 153 : it's a property of the division by 9.

Patrick C. admits that one can sometimes be misled by one's own
enthusiasm. Here in this case, a property of the reciprocal of 9
was given instead of a property of the reciprocal of 153. Don't worry
the important thing is that we try to find amazing properties for
the pleasure of everybody.

The number 153 is effectively an interesting number.
A few days ago I send to Shyam Sunder Gupta some new properties about 153 :

1.
(12345678 + 87654321) * 153 + 153 = 15300000000.
The zero is repeated 8 times.

2.
153 divides the numbers 1234567812345678 ... 12345678
(the number 12345678 is repeated 2 x n times, n positive integer).
Examples :
153 divides 1234567812345678
153 divides 12345678123456781234567812345678
153 divides 123456781234567812345678123456781234567812345678
...

3.
(1 x 5 x 3) + (1 + 5 + 3) + (1 – 5 – 3) divides 153.
(1 + 5 + 3) divides 153.
We can write :
153 = 13 + 53 + 33 = (1 + 5 + 3) * ((1 x 5 x 3) + (1 + 5 + 3) + (1 – 5 – 3)).



Co-editor Terry Trotter
A000041 Prime Curios! Prime Puzzle
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