Fun with Mathematics

Fun with Mathematics

It is Mathematics that improves our inherent skill. For example, if you are requested to find the remainder of  7²⁰⁰⁸ + 9²⁰⁰⁸ when divided by 64,you can’t follow any known conventional methods. To workout It needs intelligence. The moment you try to solve the problem, your intelligence gets boosted up. When you repeat the problem with variation, the intelligence is multiplied.

7² + 9² = 49 +  81 = 130 ; When it is divided by 64 , the remainder will be 2

7³ + 9³ = 343 + 729 =1072 ; on division, the remainder is  48.

7⁴ + 9⁴ =2401 + 6561 =8962 ; the remainder is  2

7⁵ + 9⁵ = 16807 + 59049 = 75856 ; the remainder is 16

As the power is increased the physical approach becomes cumbersome. For higher powers we have to follow possible alternative methods.

7=8-1 and 9=8+1, 64 = 8²

7⁵ + 9⁵ = (8-1)⁵ + (8+1)⁵

On expanding it as polynomials we have,

            = (8⁵ – 5x 8⁴ + 10x 8³ – 10x 8² + 5 x 8 – 1) +(8⁵ + 5x 8⁴ + 10x 8³ + 10x 8² + 5 x 8 + 1)

In both the polynomials all the terms are with some power of 8 .It is noted that all the terms will be divisible by 8² except the last two terms. For all odd n in 7ⁿ + 9ⁿ ,the last terms will cancel each other. Hence the remainder would be  the sum of last but terms in both the polynomials. It is 2 x 5 x 8= 80 on dividing by 64, the remainder will be 16. What a wonder !

For even n(n=2,4,6,….2008 and so on) , the last but one terms will cancel each other while the sum of last terms only give the remainder. It is true for all even n, whatever may its value.

If n = 2007, then remainder is obtained from 2 x 8x2007/64 =2007/4 ,reduced remainder is 3 ,original remainder 3 x 16 = 48   

When n =3 ,the remainder is obtained from 2x8x3 /64= 48

If n =2009, then the remainder is obtained from 2 x8x 2009 = 2009/4,reduced remainder is 1,original remainder is 16 and so on.

Remember the power series

(x±y)ⁿ = xⁿ ± nC₁ x^n-1 y + nC₂ x^n-2 y² ± nC₃ x^n-3y³ + ………….(-1)ⁿ y^n.

Knowing this technique, we can further explore the problem with little variation.

 Find the remainder when 7¹⁰+ 11¹⁰  is divided by 81.

(9-2)¹⁰ + (9+2)^{10 ,}  n is even ,so the remainder can be obtained from the sum of the  last terms in the polynomials. It is 2x 2¹⁰.= 2x1024, the remainder is 52