2. DEFINITION
If sum of all the factors of a number
(other than the number itself) comes out to
be the number itself, then the number is said
to be a perfect number.
3. For example
Factors of 6= 1,2,3 and 6
We observe that 6=1+2+3 (6, the number
itself is excluded)
Thus 6 is a PERFECT NUMBER.
In fact 6 is the smallest perfect number.
4. Similarly ,
Factors of 28= 1, 2, 4, 7, 14 and 28
And 28 = 1+2+4+7+14 (28 excluded)
Thus 28 is a perfect number.
5. By now, we have learned, how to check if a given
number is perfect or not.
But
How much time will you take If I ask you to list
first 10 perfect numbers?
Probably half an hour
or more…
6. Or may be much more…
Lets learn a way of finding perfect numbers
at a go…
7. Let p be a prime number
Then consider
(2p-1) and 2p-1
Then,
A number k= (2p-1) X 2p-1
Is always a perfect number.
8. Let us list a few prime numbers…
2,3,5,7,11,13,… etc
For p=2,
(2p-1) = (22-1) and 2p-1 = 22-1
= 4-1 = 21
= 3 = 2
And k = (2p-1) X 2p-1
= 3 X 2
= 6 (which is a perfect number)
Note: 6 is the smallest perfect number.
9. For p=3,
(2p-1) = (23-1) and 2p-1 = 23-1
= 8-1 = 22
= 7 = 4
And k = (2p-1) X 2p-1
= 7 X 4
= 28 (which is a perfect number)
Note: 28 is second perfect number.
10. For p=5,
(2p-1) = (25-1) and 2p-1 = 25-1
= 32-1 = 24
= 31 = 16
And k = (2p-1) X 2p-1
= 31 X 16
= 496 (which is a perfect number)
Note: 496 is third perfect number and there does not exists
any perfect number between 28 and 496.
11. Now, you try to find out
next perfect number.
Then try to find first 10
perfect numbers…
Share the trick with your
friends…