__Creative thought-9__

Health is a kind of wealth. In fact it is more valuable than the conventional wealth.

Health is required not only to our physical body but also to our mental brain. Mental health is different than that of the physical health. As it is not exhausted on continuous use, there is no need for recharging. In fact it is strengthened up every time it is used up.

The real wealth is the acquisition of profound knowledge which cannot be stolen or snatched by others.

__Recreational Mathematics__(Higher power of a number)

The fifth power of a number can be expressed as

NxNxNxNxN = (NxNxN-1)(NxNxN+1) + 1 /N

and NxNxNxNxN – N = N(NxNxNxN-1) = (N-1)N(N+1)(NxN+1)

Since 5 is a factor to (NxNxNxN-1) and 6 is a factor to (N-1)N(N+1),any three successive numbers in natural series ,NxNxNxNxN –N will have 30 as factor for all values of N. Hence the fifth power of a number can be expressed as

2x2x2x2x2 = 1x30 +2 = 15 x 2 + 2 = 5x(3x2) +2 = (2x2+1)1x2x3 + 2

3x3x3x3x3= 8x30 + 3 = 80 x3 + 3 = 20 x(4x3) +3 = (3x3x3+1)2x3x4 + 3

4x4x4x4x4= 34 x 30 + 4 = 255 x 4 + 4 = 51 x(5x4) + 4 = (4x4 +1)3x4x5 + 4

5x5x5x5x5= 104 x30 + 5 = 624 x5 + 5 = 104 x (6x5)+5 = (5x5 +1)4x5x6 + 5

The sixth power of a number N can be expressed as

NxNxNxNxNxN = NxN (NxNxNxN-1) + NxN

The term NxN (NxNxNxN-1) has a factor 60 for all values of N, hence NxNxNxNxNxN

can be shown as the sum of NxN and a multiple of 60.

2x2x2x2x2x2 = 64 = 10 x 6 +4 = 1x 60 + 4

3x3x3x3x3x3 = 729 = 120 x 6 + 9 = 12 x 60 + 9

4x4x4x4x4x4 = 4096 = 680 x 6 +16 = 68 x60 + 16

5x5x5x5x5x5 = 15625 = 2600 x6 +25 = 260 x 60 +25

It shows that NxN (NxNxNxN -1) is always divisible by 60.

Seventh power of a number

In the case of 7 th power of a number

2x2x2x2x2x2x2 = 128 = 120 + 8

3x3x3x3x3x3x3= 2187 = 18 x 120 + 27

4x4x4x4x4x4x4 = 16384 = 136 x 120 + 64

5x5x5x5x5x5x5 = 78125 = 650 x 120 + 125

Thus NxNxN (NxNxNxN-1) is always divisible by 120. In general the term NxNxNx……(k times) (NxNxNxN-1) is divisible by 2x2x2……(k-1) times x 30 for k greater than equal to l .

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