91Ó°ÊÓ

Write each of the following numbers as a product of primes \(p_{1}^{n_{1}} p_{2}^{n_{2}} \cdots p_{k}^{n k}\), where \(0

If \(a, b, c \in \mathbf{Z}^{+}\)and \(a \mid b c\), does it follow that \(a \mid b\) or \(a \mid c\) ?

For \(a, b, n \in \mathbf{Z}^{+}\), prove that \(\operatorname{gcd}(n a, n b)=n \operatorname{g} c d(a, b)\).

Find the value of the largest positive integer \(n\) such that \(2^{n}\) divides \(22 !\)

If \(a, b\) are relatively prime and \(a>b\), prove that \(\operatorname{gcd}(a-b, a+b)=1\) or 2 .

Let \(a, b \in \mathbf{Z}^{+}\)with \(a\) even and \(b\) odd. Prove that \(\operatorname{gcd}(a, b)=\operatorname{gcd}(a / 2, b)\).

Give a recursive definition for each of the following integer sequences. a) \(2,4,16,256, \ldots\) (or, \(\left.2,2^{2},\left(2^{2}\right)^{2},\left(\left(2^{2}\right)^{2}\right)^{2}, \ldots\right)\) b) \(2,4,16,65536, \ldots\) (or, \(2,2^{2}, 2^{(2)}, 2^{\left(2\left(2^{2}\right)\right)}, \ldots\) )

a) How many positive divisors are there for \(n=2^{14} 3^{9} 5^{8} 7^{10} 11^{3} 13^{5} 37^{10}\) ? b) For the divisors in part (a), how many are i) divisible by \(2^{3} 3^{4} 5^{7} 11^{2} 37^{2}\) ? ii) divisible by \(1,166,400,000\) ? iii) perfect squares? iv) perfect squares that are divisible by \(2^{2} 3^{4} 5^{2} 11^{2}\) ? v) perfect squares that are divisible by \(2^{3} 3^{4} 5^{4} 7^{5}\) ? vi) perfect cubes? vii) perfect cubes that are multiples of \(2^{10} 3^{9} 5^{2} 7^{5} 11^{2} 13^{2} 37^{2} ?\) viii) perfect fourth powers? ix) perfect fifth powers? x) perfect squares and perfect cubes?

Prove that for any \(n \in \mathbf{Z}^{+}, n>9 \Rightarrow n^{3}<2^{n}\).

For any \(n \in \mathbf{Z}^{*}\), prove that \(n\) is a perfect square if and only if \(n\) has an odd number of positive divisors.