NUMBER THEORY. 2.)show that the number N-15\" + 3n·5\"\' + 3, 5\",2 (neN) is div

Question

NUMBER THEORY.

2.)show that the number N-15" + 3n·5"' + 3, 5",2 (neN) is divisible by 31. 3.)If you know that the number 2a+3b+c+2d is divisible by 11 show that the number 13a+25b+89c+134d is also divisible by 11. 4.) Show that a six digits number with the first three digits the same and the last three digits the same is divisible by 37. 5.) Let A-30+3+3+. . +35 (i) A is an even or an odd number? (ii) A is divisible by 13? 6.) Show that 340 divide A=4+42+43. . .+42000 7.) Find ne N such that the numbers n+1, n2+3, n) +5, n.7, ns +9 are in the same time primes.

Explanation / Answer

1. We have to show that the number       N = 15n+3n . 5n+1+3n. 5n+2 is divisible by 31 for all natural number n.

We will proceed with the method of mathematical induction.

for,    consider the case n = 1

Then,                                   N = 151+31 . 51+1+31. 51+2

                                                 = 15 + 75 + 375

                                     = 465

which is divisibe by 31. ( 15 X 31 = 465).

Now assume that the result is true for a a natural number k.

i.e,                  N = 15k+3k . 5k+1+3k. 5k+2      is divisible by 31

       Thus         N = 31 C     for some natural number C .....................................(1)

To complete our proof we need to show that the result holds for n = k+1

                    N = 15k+1+3k+1 . 5k+1+1+3k+1. 5k+1+2

                           = 15 .15k +3 . 3k . 5 . 5k+1+3 . 3k. 5 . 5k+1

                             = 15 ( 15k + 3k . 5k+1 + 3k. 5k+2 )

From (1)            = 15 ( 31 C )

                        = 31(15 C)

Which is divisible by 31.

Thus by principle of mathematical induction , the result is true for every natural number n.

i.e, The number    N = 15n+3n . 5n+1+3n. 5n+2 is divisible by 31 for all natural number n.

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