Proton

STQM2103: ASSIGNMENT 2 (Due date: 13 AUGUST 2009) 1.4: DIVISION IN THE INTEGERS For question 1 and 2, for the given integers m and n, write m as qn + r, with 0 < r < n. 1) m = 64, n = 3 2) m = 48, n = 12 3) Write each integers as a product of powers of primes (as in Theorem 3). k1

k2

ks

n = p1 . p2 . … ps

a) b) c) d) e)

828 1666 1781 1125 107

For question 4 – 7, find the greatest common divisor d of the integers a and b, and write as sa + tb. 4) 5) 6) 7)

a = 60, b = 100 a = 45, b = 33 a = 34, b = 58 a = 77, b = 128

For question 8 – 11, find the least common multiple of the integers. 8) 72, 108 9) 150, 70 10) 175, 245 11) 32, 27 1.5: MATRICES  3 − 2 5 1) Let A =  ,B=  4 1 2

 3   − 2   , and C =  4 

2 3 4  5 6 − 1  . 2 0 8 

a) What is a12, a22, a23? b) What is b11, b31? c) What are the elements on the main diagonal of C?

 a + b c + d   4 6 2) If   =   , find a, b, c and d. c − d a − b  10 2  a + 2b 2a − b  4 − 2 3) If   =   , find a, b, c and d. 2c + d c − 2d  4 − 3 For question 4 – 7, let 2 1 3  A=  , B =  4 1 − 2

0 1  1 2   ,C= 2 3

1 − 2 3  − 3 2  4 2 5 , D =  4 1 , E =     0 1 2

3 2 − 1 5 4 − 3  , 0 1 2 

2 3 F=   4 5 4) If possible, compute a) C+E b) AB c) CB+F d) AB+DF 5) If possible, compute a) A(BD) b) A(C+E) c) FD+AB 6) If possible, compute a) AT and (AT)T b) (C+E) and CT+ET 7) If possible, compute a) AT+(D+F) b) (BC)T For question 8 – 10, compute A ∨ B , A ∧ B AND A Θ B . 1 0 8) A =  ,B= 0 1 

1 1 0 1  

1 1 9) A =  ,B= 1 1

0 0  1 0  

1 0 0   10) A = 0 0 1 , B = 1 0 1

1 1 1 1 1 1   1 0 0