Problem 7: show that 4^n + 4^n+1 + 4^n+2 + 4^n+3 + 4^n+4 is divisible by 11 for any positive integer

If n is even, n

4

+4

n

is divisible by 4

∴ It is composite number

If n is odd, suppose n=2p+1, where p is a positive integer

Then n

4

+4

n

=n

4

+4.4

2p

=n

4

+4(2

p

)

4

which is of the form n

4

+4b

4

, where b is a positive integer (=2

p

)

n

4

+4b

4

=(n

4

+4b

2

+4b

4

)−4b

2

=(n

2

−2b

2

)

2

−(2b)

2

=(n

2

+2b+2b

2

)(n

2

−2b+2b

2

)

We find that n

4

+4b

4

is a composite number consequently n

4

+4

n

is composite when n is odd.

Hence n

4

+4

n

is composite for all integer values of n> 1.

answer:   35 % is the probability of a late departure or an early arrival.

step-by-step explanation:

let l represents the late departure and e represents the early departure.

then, according to the question,

p(l) = 12 %,   p(e) = 27 % and = 4 %

and, we have to find out,

but, we know that,

which is the required equation that will use to find the probability of a late departure or an early departure.

therefore,   the probability of a late departure or an early arrival =   35 %