Adv Alegbra HW Solutions 36

Adv Alegbra HW Solutions 36 - 36 1.96 Let m m be positive...

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36 1.96 Let m , m 0 be positive integers, let d = ( m , m 0 ) , and let b b 0 mod d . Prove that any two solutions of the system x b mod m x b 0 mod m 0 are congruent mod ` , where ` = lcm { m , m 0 } . Solution. Absent. 1.97 On a desert island, f ve men and a monkey gather coconuts all day, then sleep. The f rst man awakens and decides to take his share. He divides the coconuts into f ve equal shares, with one coconut left over. He gives the extra one to the monkey, hides his share, and goes to sleep. Later, the second man awakens and takes his f fth from the remaining pile; he too f nds one extra and gives it to the monkey. Each of the remaining three men does likewise in turn. Find the minimum number of coconuts originally present. Solution. Here are the equations arising from the story. Let
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