Curious Properties of 153

Mathematics is something beautiful if you can see how interesting each natural number is. The moment we understand the properties of numbers, mathematics become so simple and exciting.

One such very interesting number is 153. Here are some attention-grabbing and curious properties of 153.

Property 1

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

153 = 1³ + 5³ + 3³

Because of this property 153 is called a narcissistic number (also known as an Armstrong number or a plus perfect number) i.e. a number that is cubes of each digit of the number is equal to the number itself.

Property 2

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

153 = 1+2+3+4+………..+16+17

So, it can be called a triangular number. In other words, 153 is the 17^th triangular number.

Reverse of 153, i.e. 351 is also a triangular number; hence, 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

The Sum of the digits of 153 is a perfect square-

1 + 5 + 3 = 9 = 3²

The sum of aliquot divisors of 153 is also a perfect square:

1 + 3 + 9 + 17 + 51 = 81 = 9²

Aliquot divisors of a number are all the divisors of that number excluding the number itself but including 1. It is seen that the sum of aliquot divisors of 153 is the square of the sum of the digits of 153.

Property 5

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

153 = 3 x 51

Note that the digits used in multipliers are same as in product.

Property 6

1⁰ + 5¹ + 3² = 15 = 1 * 5 * 3

Property 7

On adding the number 153 to its reverse, 504 is obtained, whose square is the smallest square which can be expressed as the product of two different non-square numbers which are reverse of one another:

153 + 351 = 504

504² = 288 x 882

Property 8

153 is divisible by the sum of its own digits:

153 / (1 + 5 + 3) = 17

Property 9

(12345678 + 87654321) * 153 + 153 = 15300000000.

The zero is repeated 8 times.

Property 10

Square root of 153 (i.e. 12.369) is the amount of full moons in one year. 

Isn’t that interesting?

Property 11

Palindromic side-effects!

A 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

However, this property is true for any number, whose reciprocal is a repeating number.

Property 12

“What’s exceptionally cool, though, is that if you take any three-digit multiple of 3, and then add the third power of its digits, and then add the third power of the digits of the result, and keep doing that, you’ll always get back to 153.”

Take 369 for instance which is a multiple of 3.

Sum of 3^rd power of the digits of 369 = 3³ + 6³ + 9³ = 27 + 216 + 729 = 972

Now, sum of 3^rd power of the digits of 972 = 9³ + 7³ + 2³ = 729 + 343 + 8 = 1080

Sum of 3^rd power of the digits of 1080 = 1³ + 0³ + 8³ + 0³ = 1 +0+ 512 + 0 = 513

Sum of 3^rd power of the digits of 513 = 5³ + 1³ + 3³ = 125 + 1 + 27 = 153