MesMath

Automorphic Numbers

If all the powers of the numbers ending with a given digit, ends with the same digit(number),then it is called an automorphic number.

Eg; 1²=1 : 11²= 121 : 21²=441…….

1³= 1 ; 1⁴ = 1 ;1^{5 =} 1 etc….

1 is an automorphic number.

Other examples are 5 and 6.

6^{2 =} 36                                                  5² = 25

6³ = 216                                               5³= 125

76² = 5776                                           25² = 625                                 

76³= 438976                                        25³ = 15625

Hurshad Numbers

If the sum of digits of a number is a factor of the number, then it is called a Hurshad Number.

Eg: 476: Sum of the digits = 4+7+6 =17.

17 is a factor of 476.

629, 782, 935, 1088 are other examples of Hurshad Numbers.

Kaprekar Numbers:

If the square of a number we make two parts and on adding  these parts we get the same number, then it is called Kaprekar Number.

Eg: 55

55² = 3025 ;  (30/25) . Here when we add :30+25 = 55.

Similarly, 45²= 2025 ; (20/25) 

Adding the parts 20 +25 = 45

45, 55 etc are Kaprekar Numbers.

Defective Numbers

The number whose sum of factors other than itself is less than itself is called a defective Number.

Eg: 10

The factors of 10 are 1, 2, 5.

Sum of these factors = 1+ 2+ 5 = 8 < 10.

Excessive Numbers

The number whose sum of factors other than itself is greater than itself is called an excessive number.

Eg:  12.

Factors of 12 are 1, 2, 3, 4, 6

Sum of these factors : 1+ 2+ 3 + 4+ 6 = 16 > 12.