1.Two numbers a and b are chosen from the set{1,2.......25}. Find the probability that a^4-b^4 is divisible by 5?
2. Two numbers x and y are chosen at random(without replacenent) from among the numbers 1,2,3,....99.Find the probability that x^3-y^3 is divisible by 3?
The answer to 1 is 2/3 and to 2 is 16/49.
The first one (assuming no repetition):
a4-b4=(a+b)(a-b)(a2+b2)
Series having 1: {(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,11).........} the pattern can be seen clearly.
No. of elements = 19
Series having 2 & beyond: {(2,3),(2,4),(2,6),(2,7),(2,8),(2,9),(2,11),......} the pattern can be seen clearly.
No. of elements = 18
Considering such series excluding 5,10,15,20,25:
No. of elements = 19+18+17+16+................+1 = 190
Now, considering elements 5,10,15,20,25
{(5,10),(5,15),(5,20),(5,25),(10,15),(10,20),(10,25),(15,20),(15,25),(20,25)}
No. of elements = 10
Total no. of favourable elements = 200
Hence, probability = \(\frac{200}{C(25,2)}\)
Now, you can try the second one and if you still have problem with that, you can send me private message in virtual classroom and I can schedule a short live online session for that.