Interesting Properties of 153

Posted: 13 May 2011 12:26 AM PDT

Mathematics is something beautiful if you can see how interesting each natural number is. All numbers are interesting, but some numbers are more interesting than others.

In my earlier posts I have discussed about such interesting numbers –

Palindromes , Munchausen Number, beauty of numbers, Ramanujan Number

One such very interesting number is 153. I figured out few properties of 153 myself and felt proud on my observations. However when I researched more I was embarrassed at the paucity of my observations. I swear now that each number is a study in itself.

I am listing below some attention-grabbing and curious property of 153. For better understanding, I have linked certain terms to wikipedia-

Properties of 153

Property 1

153 is the smallest number which can be expressed as the sum of cubes of its digits –

153 = 1^3 + 5^3 + 3^3

Because of this property 153 is called a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number or a plus perfect number) is a number that is the sum of its own digits each raised to the power of the number of digits. Source: wikipedia

Property 2

153 can be expressed as the sum of all integers from 1 to 17.

153 = 1+2+3+4+………..+16+17

Hence it can also be called a triangular number. Reverse of 153, i.e. 351 is also a triangular number, 153 can be termed as a reversible triangular number.

Property 3

153 is equal to the sum of factorials of number from 1 to 5 –

153 = 1! + 2! + 3! + 4! + 5!

Property 4

Sum of the digits of 153 is a perfect square-

1 + 5 + 3 = 9 = 3^2

Sum of all the divisors of 153 (except itself) is also a perfect square-

1 + 3 + 9 + 17 + 51 = 81 = 9^2

Property 5

153 can be expressed as the product of two numbers formed from its own digits-

153 = 3 x 51

Property 6

Square root of 153 = 12.369 is the amount of full moons in one year. Isn't that interesting?

Other Observations on 153 –

1^0 + 5^1 + 3^2 = 15

1^1 + 5^2 + 3^3 = 53

153 if added to its reverse351,we get 504.

153 + 351 = 504

Square of 504 is the smallest square which can be expressed as the product of two different non-square numbers which are reverse of one another:

504^2 = 288 x 882

I hope now you find the number 153 as interesting as I did. I have purposefully not mentioned few other interesting properties of 153. Now the ball is in your court. Share more such properties with other readers by posting a comment below.

Suggested by Mohit Singhvi

Written by Vineet Patawari