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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
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.
country uses more than 3 letters to make a word. What is the highest number of
words one can make in that language?
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:
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.
GREATEST COMMON FACTOR (GCF) AND THE LEAST COMMON MULTIPLE (LCM)
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.
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.