SOLUTION: This is a question my teacher assigned if we want to do, it is extra, but I would like to learn more and this is a great problem. I cannot understand it no matter how hard I try. P

Question 1046020: This is a question my teacher assigned if we want to do, it is extra, but I would like to learn more and this is a great problem. I cannot understand it no matter how hard I try. Please explain fully and how you got your answer.

For any positive integer, find the sum of the squares of its digits. If the result is 1, you�re done. Otherwise, find the sum of the squares of the digits of the result. Continue this process until the result is 1, in which case we call the original integer a happy number, or until there is a repeated result, which leads to an endless loop, in which case we call the original integer an unhappy number. For example, consider 13 and 85.

13 is a happy number since
12 + 32 = 1 + 9 = 10,
12 + 02 = 1 + 0 = 1.
85 is an unhappy number since
82 + 52 = 64 + 25 = 89,
82 + 92 = 64 + 81 = 145,
12 + 42 + 52 = 1 + 16 + 25 = 42,
42 + 22 = 16 + 4 = 20,
22 + 02 = 4, 42 = 16,
12 + 62 = 1 + 36 = 37,
32 + 72 = 9 + 49 = 58,
52 + 82 = 25 + 64 = 89, which is a repeated result.
What is the least unhappy number?

What is the least prime number that is a happy number?

What is the least integer greater than 1 that, when multiplied by any happy number, yields another happy number?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Squares end in 0, 1, 4, 5, 6, or 9.
To add up to 1 (or 10, or 100), only a few combinations work.
For a 2-digit numbers, you need to pair
0 with 1,
or 1 with 3, or
6 with 8.
Obviously is a happy number, and so are , and .
Other happy numbers are
, , and .
The least unhappy number is .

2^2+0^2=4}}} and we went full loop back to .
That also tells us that as soon as we found a number, or a sum of squares that is
, or , or , or , or , or , or , or ,
we know we have entered the loop above,
meaning that the original number is an unhappy number,
and so are all the other intermediate results.
Numbers with the same digits in a different order, are also unhappy,
because they will yield the same sum of squares.
The least prime that is a happy number is obviously not .
It is not either, because

and is as unhappy as ,
bringing us into the same -->37-->58-->89--->145... unhappy loop we found for , is also unhappy:

, and , as unhappy as brings us back into the
--->89--->145 ... loop we knew from .
The next prime is happy number :

, and as
is happy, making
, is the least happy prime number.From the sums above,m we see that and are also happy numbers.
Obviously is a number greater than ,
than when multiplied by any happy number gives a product with the same digits as the original number plus a zero added at the end,
so the product is also a happy number.
If there were a smaller factor whose products times a all happy numbers were happy, it would be a happy single digit number itself (as a product time happy .
However, all digits other than and are unhappy, and while is happy is not:

, and we knew that is unhappy from testing .
So, I think the least nteger greater than 1 that, when multiplied by any happy number, yields another happy number is .
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