Combinatorics | Number Theory Answers

A pay-off matrix can have more than one saddle point. True or false

In a transportation problem, if all the source availabilities a_i and all the requirements
b_j are integers, then the optimal solution consists of integers only.
Its True or false ?

1.Show that if d ≠ 0, then d | (-a) and -d | a.
2.Show that it is false that a>b implies a|b.
3.Is 980637 divisible by 7? Show.
4.Determine whether of the following are divisible by 3, 5, 7, 9, or 11 using the methods described int he text:
A. 1969
B. 28350
C. 1421
D. 17303
E. 116424
F. 1089
5.Classify each of the following as true or false:
A. 6 is a divisor of 24.
B. 40 is a multiple of 8.
C. 0 divides 10.
D. 13 is a factor of 33.
E. 12 divides 6.
6.Show that 23n -1 is divisible by 7.
7.Show that 5n - 1 is divisible by 4.

If there are 3 roads from town A to town B And 4 roads from town B to town C, in how many ways can one go from town A to town C and back to town A, through town B, without passing through the same road twice?

Q. prove that √n is irrational.

A panel is conducting an interview on six candidates of different heights. If they are to put them in line, in how many ways can they arrange them in line such that no three consecutive candidates are in increasing order of height from front to back?

What is the total number of outcomes of choosing between water, iced tea, or lemonade; with a lemon slice or without; served in a small or large glass?

Suppose a prime can be written as p = x^2 + 5y^2. Show that
p ≡ 1 or 9 (mod 20). Assume that p > 5

There are 15 boys and 10 girls in your class. A four member committee is to be formed from the students
of your class. In how many ways this can be done if the committee consists at least three girls?

There are 15 boys and 10 girl in your class. A four number committee is to be formed from the student of your class. In how many ways this can be done if the committee consist at least three girls?