Interesting Properties of Numbers

Here are some interesting properties of Number:

Square of Sum = Sum of Cubes

An interesting property of numbers is that, the square of sum of first ‘n’ natural numbers is equal to the sum of their cubes:

(1 + 2 + 3 +...+ n)² = 1³+ 2³+ 3³+...+ n³

Examples:

(1 + 2 + 3 + 4 + 5)² = 225 = 1³ + 2³ + 3³ + 4³ + 5³

(1 + 2 + 3 + 4 + 5 + 6)² = 441 = 1³+2³+3³+4³+5³+6³

(1+2+3+4+5+6+7)² = 784 = 1³+2³+3³+4³+5³+6³+7³

Sum of all the digits of a Cube

The sum of any cube when reduced to a single digit will always be either 1, 8, or 9.

For Example:

5³ = 125 = 1+2+5 =8

9³ = 729 = 7+2+9 = 18=1+8=9

17³ = 4913=4+9+1+3=17=1+7=8

4³=64=6+4=10=1+0=1

also, they follow a cycle. The numbers 1, 8, and 9 appear in order:

1 - sum of the digits of cube of 1 would be 1

2 - would be 8

3 - would be 9

4 - would be 1 (4³ = 64 = 6+4 = 10= 1+0 = 1)

5 - would be 8 (5³ = 125 = 1+2+5 = 8)

6 - would be 9 (6³ = 216 = 2+1+6 = 9)

So on and so forth.

Armstrong Number

If sum of cubes of each digit of the number is equal to the number itself, then the number is called an Armstrong number.

For example,

153 = 1³ + 5³ + 3³

370 = 3³ + 7³ + 0³

371 = 3³ + 7³ + 1³

407 = 4³ + 0³ + 7³

Parasitic Number

A Parasitic number (in base 10) is a Positive natural number which can be multiplied by n by moving the rightmost digit of its decimal representation to the front.

102564 × 4 = 410256

142857 × 5 = 714285

179487 × 4 = 717948

105263157894736842 × 2 = 210526315789473684

1014492753623188405797×7=7101449275362318840579

Munchausen Number

A Munchausen Number is a number that is equal to the sum of its digits each raised to a power equal to the digit. It is also called perfect digit-to-digit invariant (PDDI) because of its feature.

The only Munchausen numbers are 1 and 3435.

Specialty of 3435

3³ + 4⁴ + 3³+ 5⁵ = 27 + 256 + 27 + 3125 = 3435

though, 0⁰is undefined but if we consider it as 0, then there is one more Munchausen number founded till date - that number is 438579088.