Question

7. Find the probability that the difference in squares 2 -y2 of two random integers z, y is divisible by 3. ( (Hint: An integer is divisible by 3 iff the remainder of division by 3 is 0. Observe that a-y and (+3t)-(y+3s) are both divisible by 3 at the same time, no matter s,t the function f(r,y) Z. Thus, to build the sample space, it suffices to consider [2,2-v2],-x2-y2 mod 3 for x, y e z,-10]s, [1], [2]a), i.e., mod 3)

Answer 1

given that x^2-y^2 should be diivsible by 3

x=3k ,3k+1,3k+2

y=3t, 3t+1,3t+2

which implies

x^2 is of the form

3m ,3m+1 from squaring 3k ,3k+1,3k+2

similarly

y^2 is the of form 3n , 3n+1

from squaring y=3t, 3t+1,3t+2

so

3m,3m+1 and 3n ,3n+1

total number of choosings are 4 (x^2,y^2)=(3m,3n) ,(3m,3n+1),(3m+1,3n) ,(3m+1,3n+1)

which are divisibe by 3 are (3m,3n) ,(3m+1,3n+1)

hence the probability is 2/4 = 1/2 =0.5