#8 THE THEORY OF CONGRUENCES837 (a) Obtain three consecutive integers, each having a square factor.[Hint: Find an integer a such that 22 | a, 32 | a + 1, 5² | a + 2.](6) Obtain three consecutive integers, the first of which is divisible by a square, thesecond by a cube, and the third by a fourth power.8. (Brahmagupta, 7th century A.D.) When eggs in a basket are removed 2, 3, 4, 5, 6 at atime there remain, respectively, 1, 2, 3, 4, 5 eggs. When they are taken out 7 at a time,none are left over. Find the smallest number of eggs that could have been contained inthe basket.9. The basket-of-eggs problem is often phrased in the following form: One egg remainswhen the eggs are removed from the basket 2, 3, 4, 5, or 6 at a time; but, no eggs remainif they are removed 7 at a time. Find the smallest number of eggs that could have beenin the basket.10. (Ancient Chinese Problem.) A band of 17 pirates stole a sack of gold coins. When theytried to divide the fortune into equal portions, 3 coins remained. In the ensuing brawl overwho should get the extra coins, one pirate was killed. The wealth was redistributed, butthis time an equal division left 10 coins. Again an argument developed in which anotherpirate was killed. But now the total fortune was evenly distributed among the survivors.What was the least number of coins that could have been stolen?11. Prove that the congruencesX = a (mod n)andX = b (mod m)admit a simultaneous solution if and only if gcd(n, m)|a – b; if a solution exists, confirmthat it is unique modulo lcm(n, m).12. Use Problem 11 to show that the following system does not possess a solution:andx = 7 (mod 15)(9 pou) s = x13. If x = a (mod n), prove that either x = a (mod 2n) or x = a +n (mod 2n).14. A certain integer between 1 and 1200 leaves the remainders 1, 2, 6 when divided by 9,11, 13, respectively. What is the integer?15. (a) Find an integer having the remainders 1, 2, 5, 5 when divided by 2, 3, 6, 12, respec-tively. (Yih-hing, died 717).(b) Find an integer having the remainders 2, 3,4, 5 when divided by 3, 4, 5, 6, respectively.(Bhaskara, born 1114).c) Find an integer having the remainders 3, 11, 15 when divided by 10, 13, 17, respec-

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We use basic module formula here According to the question and then take lcm of all the possible…