Combinatorics | Number Theory Answers

Statement: “The negative of any even integer is even.”
Using direct proof method

When dividing each of the numbers 3187 and 3319 by some positive integer number, the remainder happened to be equal to 8. What is the smallest possible value of the divisor?

36 students are members of a sports club. Every two of them are either friends or enemies.
(Friendship and enmity are reciprocal, i.e. if A is a friend to B then B is a friend to A, and the
same applies to being enemies.) It has turned out that each of the students has exactly 8 enemies. Let us called a group of three students concurrent if they are either pairwise enemies or pairwise friends to each other. What is the maximum possible quantity of concurrent student
triples in this sports club? (Two distinct concurrent student triples may have mutual students in
them.)

The Line A bus arrives at the bus stop every 25 minutes and the Line B bus arrives every 15 minutes. They are both at the bus stop right now. In how many minutes will they both be at the same stop again?

The sum of the first two digits of a three digit number is 4. The sum of the numbers
that is formed by using each digit of the number once is 1998. If the number has 8
factors, then what is the number?

find the smallest positive solution of the system of linear congruences ? X=2 (mod3) , X=3(mod 5), X=2(mod 7)

I have 52 cards (ordinary deck from 2 to Aces, each having four ranks ). I start picking a random card from the deck and add it to the sequence. I stop when the currently selected card belongs to the same rank as the previously selected card. No replacement is allowed. I would like to find what are the probabilities: Selecting the 5th card would be the same rank as the 4th. Or Selecting the 19th card would be the same rank as the 18th. Or None of the selected cards was the same rank as the previously selected card.I know there are 52! possible sequences that can be generated, but the tricky part would be numerator because I do not see any clear pattern.Consider the following sequence:
[ Ace, King, 7, 7 ... n=52] We could say the Probability of the 4th card belongs to the same rank as third is (52 * 48 * 47 * 3)/52!
However:
[7, King, 7,7 .. n=52]. The probability of the 4th card belongs to the same rank as third is (52*48*47*2)/52!
So we can not generalize. Or can we?

The number of ways in which 4 women and 10 men are to be seated in a row so that exactly 3 men sit between every 2 nearest women is P. Then the value of P/10! is equal to

(a) Prove the identity
1*1!+2*2!+3*3!+....+n*n!+(n+1)!-1
(b) discuss the combinatorial significance of this identity
(c)show that integer m can be expressed uniquely in the following form(factorial representation)
m=a1*1!+a2*2!+.......+ai*iI!+..
Where 0 less than or equal to ai less than or equal to i for i=1,2,3,4,.....