Perfect Numbers I

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A factor of a number is an exact divisor of that number. Now in other words we can say , a factor of a number is the number which completely divides the given number without any remainder. Let us take some examples to understand factors.

Let us factor a number 12, factor of 12 are 1,2,3,4,6 and 12. However, none of the numbers 5,7,8,9,10 and 11 is a factor of 12.Now what are Perfect numbers? if the sum of all the factors of a number is two times the number, then the number is called Perfect number. The factors of 6 are 1,2,3 and 6.Also, 1+2+3+6=12=2*6 i.e sum of all factors of 6 = 2*6 so, 6 is a perfect number. Now let us see if 28 is the perfect number .The factors of 28 are 1,2,4,7,14 and 28.Sum of all factors of 28 = 1+2+4+7+14+28=56+2*28 So,28 is also a perfect number. Note that 12 is not a perfect number.

Procedure for finding factors
Step 1 Obtain the number whose factors are to be listed.
Step2 Divide the number by 2.If the number is divisible by 2, find the quotient and list 2 and other factor (quotient) just below 2.
Step3 Divide the number by 3.If the number is divisible ...
... by 3, find the quotient and list 3 and the quotient just below 3.
Step4 Divide the number by 4.If the number is divisible by 4, find the quotient and list 4 and the quotient just below 4
Continue this process with 5,6,7 etc. till you get a factor which has already been listed.

Let us take some example of factor, Find factors of a numbers 24:-
(a) 24 The given number is 24. Clearly, 1 and 24 are its factors. So, we list 1 and 24.We find that if we divide 24 by 2, the quotient is 12 and the remainder is zero. Therefore, 24 is divisible by 2. S, we list 2 and 12.When 24 is divided by 3, the quotient is 8 and the remainder is zero.
So,24 is divisible by 3 and we list 3 and 8.Now, we divide 24 by 4. Clearly, when we divide 24 by 4, we obtain 6 as the quotient and the remainder as zero. So, 24 is also divisible by 4 and we list 4 and 6.Clearly,24 is not divisible by 5. The next number is 6, which has already been listed as a factor.

So, we stop the procedure and list the
factors of 24 as given below:
1 2 3 4
24 12 8 6
(b) The given number is 64.To factor a number clearly; it is divisible by 1,2,4,8 etc. So, its factors or Factor Numbers are as listed below.1 2 4 8 64 32 16 8
(c) The given number is 144. Clearly ,it is divisible by 1,2,4,6,8,12,16,24,etc.So,factors of 144 are:1 2 3 4 6 8 9 12 144 72 48 36 24 18 16 12

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