1. If the sum of the factors of a number is equal to that number then it is called a perfect number
2. For example 6 Is a perfect number because factors of 6 are 3, 2, 1 and 3+2+1 = 6.
3. 28 is a perfect number because factors of 28 are 14, 7, 4, 2, 1 and 14+7+4+2+1 = 28.
4. Find out the magic numbers less than 100000
5. FYI there is only one magic number between 100-1000 and only one magic number
between 1000 – 10000 and no magic numbers between 10000-100000).

Some interesting facts :

1. If 2exp(n)-1 is prime, then 2exp(n-1)*(2exp(n)-1) is a perfect number
2. All perfect numbers are even and in the form 2exp(n-1)*(2exp(n)-1).
3. There are no odd perfect numbers. If you find one file a patent
4. Any odd perfect number must exceed 10exp(300) and must be divisible by a prime power exceeding 10exp(20)
5. Perfect numbers are infinite. If you can prove the finiteness file a patent

Solution: Please ping me at: Gopala.Krishna@iitbombay.org (or)Gopala.Krishna@outlook.com

1. Take a number. Call it NumberOne.
2. Reverse all its digits. Call it NumberTwo.
3. If NumberTwo is divisible by NumberOne then they are reverse digit divisible numbers.
4. For ex: 721 reverse: 127. 721 is divisible by 127.
5. Find out this kind of unique numbers.

Solution: Please ping me at: Gopala.Krishna@iitbombay.org (or)Gopala.Krishna@outlook.com

1. There are two persons A and B
2. A knows the product of two numbers
3. B know the sum of the same two numbers
4. The numbers are between 2- 99

They started talking to each other
4 sentences exchanged between them
Based on the conversation we need to find the numbers

The conversation is:

1st: A says to B that he don't know the two numbers
2nd: B replies that i know u don't know the two numbers
3rd: A then replies NOW i know the two numbers
4th: B then replies NOW I ALSO know the two numbers

Solution: Please ping me at: Gopala.Krishna@iitbombay.org (or)Gopala.Krishna@outlook.com

1. If a number is a square of an integer, and also the difference between cubes of two consecutive integers,
then the square root of this number can be expressed as a sum of squares of two consecutive integers.
2. For e.g. 169 = 512 - 343, so 13 = 4 + 9.
3. Find this kind of unique numbers.

Solution: Please ping me at: Gopala.Krishna@iitbombay.org (or)Gopala.Krishna@outlook.com

1. Find the numbers which can be expressed in two ways as sum of two cubes
2. For eg. 1729 = 1 cube + 12 cube = 9 cube + 10 cube (this is the least numer called ramanujam's number)
3. 1729 can also be expressed as product of three prime numbers i.e. 1729 = 7 * 13 * 19
4. Another such number is 885623890831 = 7511 cube + 7730 cube = 8759 cube + 5978 cube
5. 885623890831 can also be expressed as 885623890831 = 3943 * 14737 * 15241
6. Find this kind of unique numbers

Solution: Please ping me at: Gopala.Krishna@iitbombay.org (or)Gopala.Krishna@outlook.com