Combinatorics | Number Theory Answers

a) i) Give an inductive formula for the sum of the first n odd numbers:

1 + 3 + 5 + ... + 2n -1

Show your induction process.

ii) Use the proof by mathematical induction to prove the correctness of your

inductive formula in i) above.

Show that 2²⁴-1 and 2¹⁶-1 are composite. Hints:use expansion of (a²-b²)

Show that any prime number is either in form of 4k+1 or 4k+3, k is any positive integer

Six professor begin courses on Monday, Tuesday, Wednesday, Thursday, friday, and Saturday, and announce their intention of lecturing at interval of 3,2,5,6,1and 4 days respectively the regulations of the university forbid Sunday a lecture(so that a Sunday a lecture must be omitted) when first will all six professors find themselves compelled to omit a lecture

Six professor begin courses on Monday, Tuesday, Wednesday, Thursday, friday, and Saturday, and announce their intention of lecturing at interval of 3,2,5,6,1and 4 days respectively the regulations of the university forbid Sunday a lecture(so that a Sunday a lecture must be omitted) when first will all six professors find themselves compelled to omit a lecture

Suppose f : N → N is a function defined by ƒ(x) = a(x + b), where a, b E N. If f is a bijective function,

then find the value of a + b.

Suppose f : N → N is a function defined by ƒ(x) = a(x + b), where a, b E N. If f is a bijective function,

then find the value of a + b.

Let us define a function f : N → Q as follows,

(n-1)/4 n is odd

f(n) = {

(n+1)/2 n is even

Which of the following are true?

f is not one to one.

f is not onto.

f is onto.

f is one to one.

In a Zoo, there are 6 Bengal white tigers and 7 Bengal royal tigers. Out of these tigers, 5 are males and 10 are either Bengal royal tigers or males. Find the number of female Bengal white tigers in the Zoo.