Properties of the number 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 **17**^{th} 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** = 1^{3} + 5^{3} + 3^{3} = (1 + 5 + 3) * ((1 x 5 x 3) + (1 + 5 + 3) + (1 – 5 – 3)).