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Use the fact that nCk equals n!/k!(n-k)! to express in factorials
1) The coefficient "u" of x^n in the expansion of (1+x)^2n
2) the coefficient "v" of x^n in the expansion of (1+x)^2n-1, Hence show that u=2v
Find the coefficient of x^6 in the expansion of (1-3x)(1+2x)^9 using the general term: t= nCr a^n-r b^r

1. Use mathematical induction to show that 8 │ (5^2n + 7).
Hint: 5^2(k+1) + 7 = 5^2(5^2k + 7) +(7 - 5^2·7)

2. Use the Division Algorithm to establish that 3a^2 – 1 is never a perfect square.

3. Use the Euclidean Algorithm to obtain integers x and y satisfying
gcd(1769,2378) = 1769x + 2378y.

4. Determine all solutions in the positive integers of the following Diophantine equation:
123x + 360y = 99.

5. Find the prime factorization of integer 1234, 10140, and 36000.

6. Give an example of a^2 ≡ b^2 (mod n) need not imply a ≡ b (mod n).

7. Show the following statements are true:
a. For any integer a, the unit digit of a^2 is 0, 1, 4, 5, 6 or 9.
b. Any one of the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 can occur as the units digit of a^3.
c. For any inter a, the units digit of a^4 is 0, 1, 5 or 6.

The number of letters in the language of a weird country is 5 and no one in that
country uses more than 3 letters to make a word. What is the highest number of
words one can make in that language?
An optimization model includes a chance constraint to satisfy demand of a particular product. The demand is uncertain and is modeled with an integer uniform distribution with parameter value of 0 and 4. That is, the probability that the demand is 0, 1, 2, 3, or 4 is exactly the same. A decision is made to order 2 units of the product from a supplier in order to satisfy the uncertain demand. What is the value at risk (VaR) for the demand constraint?
Prove that $\sum_{i=1}^s a_i=2^{n}-1-\sum_{i=s+1}^n a_i=2^s-1. Note that $\sum_{i=1}^n=2^n-1.

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.


1.Calculate (3141 , 1592).
2.Find x and y such that, 3141x + 1592y =1.
3.Find the solution of 803x + 154y = 33.
4.Find the GCD and LCM of the numbers 63,24, 99.

Give an example of a LPP with more than one optimal solution
A company produces their products P,Q and R from raw materials A,B and C .
To produce one unit of the product P, 2 units of A, 5 units of B and 4 units of C are
required. To produce one unit of the product Q, 1 unit of A, 1 unit of B and 2 units
of C are required. To produce one unit of the product R, 1 unit of A, 1 unit of B and ,
1 unit of C are required. Profits per unit of the products P,Q and R are Rs.10, Rs 5
and Rs. 4 respectively. The company has 10 units of A, 20 units of B and 20 units of
C. Formulate the problem of maximization of profit as a LPP.
A pay-off matrix can have more than one saddle point. True or false