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Chapter 4: Problem 15
Determine those values of \(c \in \mathbf{Z}^{+}, 10
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If \(p\) is any prime, prove that \(\sqrt[3]{p}\) is irrational.
Consider the following six equations. $$ \begin{array}{r} 1) \\ 2) \\ 3) \\ 4) \\ 5) \\ 6) \end{array} $$$$ \begin{aligned} 1 &=1 \\ 1-4 &=-(1+2) \\ 1-4+9 &=1+2+3 \\ 1-4+9-16 &=-(1+2+3+4) \\ 1-4+9-16+25 &=1+2+3+4+5 \\ 1-4+9-16+25-36 &=-(1+2+3+4+5+6) \end{aligned} $$ Conjecture the general formula suggested by these six equations, and prove your conjecture.
a) Ten students enter a locker room that contains 10 lockers. The first
student opens all the lockers. The second student changes the status (from
closed to open, or vice versa) of every other locker, starting with the second
locker. The third student then changes the status of every third locker,
starting at the third locker. In general, for \(1
Let \(m, n \in \mathbf{Z}^{+}\)with \(m n=2^{4} 3^{4} 5^{3} 7^{1} 11^{3} 13^{1}\). If \(\operatorname{lcm}(m, n)=2^{2} 3^{3} 5^{2} 7^{1} 11^{2} 13^{1}\), what is \(\operatorname{gcd}(m, n) ?\)
a) How many positive integers can we express as a product of nine primes (repetitions allowed and order not relevant) where the primes may be chosen from \(\\{2,3,5,7,11\\}\) ? b) How many of the positive integers in part (a) have at least one occurrence of each of the five primes? c) How many of the results in part (a) are divisible by 4 ? How many of the results in part (b)?
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