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Happy Number

Happy number is a positive integer that gives the final number as 1 when certain steps (square and sum the digits) are performed continuously.

Checking a Happy Number

Step 1: Consider any number.

Step 2: Square the number and replace the resultant with the sum of the squares of its digits.

Step 3: Repeat the steps. The final number will be 1, or the steps will loop endlessly in a cycle.

Those numbers (considered in Step 1) that give the final number as 1 are Happy numbers.

Examples

Example 1

Step 1: Consider number 1.

Step 2: Square the number => 12 => 1 x 1 => 1.

Step 3: Repeat the steps.

12 => 1 x 1 => 1

Repeating the steps gives the final number as 1. Therefore, the number considered in Step 1 is a Happy number.

Example 2

Step 1: Consider number 2.

Step 2: Square the number => 22 => 2 x 2 => 4.

Step 3: Repeat the steps.

42 => 4 x 4 => 16

12 + 62 => 1 + 36 => 37

32 + 72 => 9 + 49 => 58

52 + 8 => 25 + 64 => 89

82 + 92 => 64 + 81 => 145

12 + 42 + 52 => 1 + 16 + 25 => 42

42 + 22 => 16 + 4 => 20

22 + 02 => 4 + 0 => 4

42 => 16

12 + 62 => 1 + 36 => 37

Repeating the steps does not give the final number as 1 and loop endlessly in a cycle. Therefore, the number considered in Step 1 is not a Happy number.

Example 3

Step 1: Consider number 3.

Step 2: Square the number => 32 => 3 x 3 => 9.

Step 3: Repeat the steps.

92 => 9 x 9 => 81

82 + 12 => 64 + 1 => 65

62 + 52 => 36 + 25 => 61

62 + 12 => 36 + 1 => 37

32 + 72 => 9 + 49 => 58

52 + 82 => 25 + 64 => 89

82 + 92 => 64 + 81 => 145

12 + 42 + 52 => 1 + 16 + 25 => 42

42 + 22 => 16 + 4 => 20

22 + 02 => 4 + 0 => 4

42 => 16

12 + 62 => 1 + 36 => 37

32 + 72 => 9 + 49 => 58

Repeating the steps does not give the final number as 1 and loop endlessly in a cycle. Therefore, the number considered in Step 1 is not a Happy number.

Example 4

Step 1: Consider number 4.

Step 2: Square the number => 42 => 4 x 4 => 16.

Step 3: Repeat the steps.

162 => 16 x 16 => 256

22 + 52 + 62 => 4 + 25 + 36 => 65

62 + 52 => 36 + 25 => 61

62 + 12 => 36 + 1 => 37

32 + 72 => 9 + 49 => 58

52 + 82 => 25 + 64 => 89

82 + 92 => 64 + 81 => 145

12 + 42 + 52 => 1 + 16 + 25 => 42

42 + 22 => 16 + 4 => 20

22 + 02 => 4 + 0 => 4

42 => 16

162 => 16 x 16 => 256

22 + 52 + 62 => 4 + 25 + 36 => 65

Repeating the steps does not give the final number as 1 and loop endlessly in a cycle. Therefore, the number considered in Step 1 is not a Happy number.

Example 5

Step 1: Consider number 5.

Step 2: Square the number => 52 => 5 x 5 => 25.

Step 3: Repeat the steps.

252 => 25 x 25 => 625

62 + 22 + 52 => 36 + 4 + 25 => 65

62 + 52 => 36 + 25 => 61

62 + 12 => 36 + 1 => 37

32 + 72 => 9 + 49 => 58

52 + 82 => 25 + 64 => 89

82 + 92 => 64 + 81 => 145

12 + 42 + 52 => 1 + 16 + 25 => 42

42 + 22 => 16 + 4 => 20

22 + 02 => 4 + 0 => 4

42 => 16

12 + 62 => 1 + 36 => 37

32 + 72 => 9 + 49 => 58

Repeating the steps does not give the final number as 1 and loop endlessly in a cycle. Therefore, the number considered in Step 1 is not a Happy number.

Example 6

Step 1: Consider number 6.

Step 2: Square the number => 62 => 6 x 6 => 36.

Step 3: Repeat the steps.

362 => 36 x 36 => 1296

12 + 22 + 92 + 62 => 1 + 4 + 81 + 36 => 122

12 + 22 + 22 => 1 + 4 + 4 => 9

92 => 81

82 + 12 => 64 + 1 => 65

62 + 52 => 36 + 25 => 61

62 + 12 => 36 + 1 => 37

32 + 72 => 9 + 49 => 58

52 + 82 => 25 + 64 => 89

82 + 92 => 64 + 81 => 145

12 + 42 + 52 => 1 + 16 + 25 => 42

42 + 22 => 16 + 4 => 20

22 + 02 => 4 + 0 => 4

42 => 16

12 + 62 => 1 + 36 => 37

32 + 72 => 9 + 49 => 58

Repeating the steps does not give the final number as 1 and loop endlessly in a cycle. Therefore, the number considered in Step 1 is not a Happy number.

Example 7

Step 1: Consider number 7.

Step 2: Square the number => 72 => 7 x 7 => 49.

Step 3: Repeat the steps.

42 + 92 => 16 + 81 => 97

92 + 72 => 81 + 49 => 130

12 + 32 + 02 => 1 + 9 + 0 => 10

12 + 02 => 1 + 0 => 1

Repeating the steps gives the final number as 1. Therefore, the number considered in Step 1 is a Happy number.

Hope you have understood this mazing concept of the Happy number. To check whether a number is a Happy number or not, use the above examples. In case of any query, please feel free to provide your valuable comments.

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