Q1. Find a if
 a ≡ 2 (mod 5)
 a ≡ 3 (mod 13)

Q2. Find a solution to 13x = 1 (mod 70)

Q3. Use the Chinese Remainder Theorem to find all solutions in such that
x = 3 mod 4
x = 2 mod 3
x = 4 mod 5:

Q4. Given x = 6 mod 13 and x = 2 mod 17, find x

Q5. X = 1 (mod 2)
X = 2 (mod 3)
X = 3 (mod 5)
X = 1 (mod 7):

Q6. Find all the integers a such that:
i. a mod 3 = 2
ii. a mod 11 = 9
iii. a mod 20 = 5

Q7. Five pirates and a monkey are shipwrecked on an island. The pirates have collected a pile of coconuts which they plan to divide equally among themselves the next morning. Not trusting the others, one pirate wakes up during the night and divides the coconuts into five equal parts with one left over, which he gives to the monkey. The pirate then hides his portion of the pile. During the night, each of the other pirates does exactly the same thing by dividing the pile he finds into five equal parts leaving one coconut for the monkey and hiding his portion. In the morning, the pirates gather and split the remaining pile of coconuts into five equal parts and again one is left over for the monkey. What is the smallest number of coconuts the pirates could have collected for their original pile?

Q8. A girl was carrying a basket of eggs, and a man driving a horse hit the basket and broke all the eggs. Wishing to pay for the damage, he asked the girl how many eggs there were. The girl said she did not know, but she remembered that when she counted them by twos, there was one left over; when she counted them by threes, there were two left over; when she counted them by fours, there were three left over; when she counted them by fives, there were four left; and when she counted them by sixes, there were five left over. Finally, when she counted them by sevens, there were none left over. `Well,' said the man, `I can tell you how many you had.' What was his answer?

Q9. Three sailors pick up a number of coconuts, place them in a pile and retire for the night. During the night, the first sailor—wanting to make  sure that he gets his fair share—gets up and takes 1/3 of the pile. The number of coconuts in the pile is not divisible by 3, there is 1 left over and he gives that coconut to the monkey.
A little later, the second sailor gets up to do the same thing. He too finds that in order to take 1/3 of the pile, he needs to give one coconut to the monkey. Even later still, the third sailor gets up and does the same thing, giving 1 coconut to the monkey. In the morning the sailors gather to divide the remaining pile of coconuts evenly among the three of them. None would dare say anything about the size of the pile for fear of incriminating himself, and the monkey isn’t talking, since he got 3 coconuts last night. When they divide the pile into 3 equal piles they find that they need to give the monkey 1 more coconut. What is the smallest number of coconuts with which they could have started and how many did each sailor get?

Other Similar Questions

Q10. A no divided by 7 leaves 4 as rem..n when divided by 11 leaves 6 as rem..wt will b remain when it is div by 13.

from post http://www.cat4mba.com/node/4912  by Chandan