Assignment # 3-Number Theory
(1) Solve the simultaneous congruence:
x = 7 (mod 9)
x =0 (mod 10)
x = 3 (mod 7)
(2) (Bhaskara) There are n eggs in a basket. If eggs are removed from the basket 2; 3; 4; 5;
and 6 at a time, there remain 1; 2; 3; 4; and 5 eggs in the basket respectively. If eggs
are removed from the basket 7 at a time, no eggs remain in the basket. What is the
smallest possible number of eggs the basket could have contained?
(3) (Ch’in Chiu-Shao) Three farmers divide equally the rice that they have grown. One
goes to a market where an 83-pound weight is used, another to a market that uses a
110-pound weight, and the third to a market using a 135-pound weight. Each farmer
sells as many full measures as possible, and when the three return home, the rst
has 32 pounds of rice left, the second 70 pounds, and the third 30 pounds. Find the
smallest possible amount of rice they took to the market.
(4) Solve the polynomial congruence x3 + 5×2 + 2x 1 = 0 (mod 72).
(5) Solve the polynomial congruence x3 + 30×2 + 27x + 23 = 0 (mod 45).
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