91Ó°ÊÓ

Let \(a, b, c,\) and \(n\) be any positive integers and \(p\) be any prime. Prove each. Let \(d=\operatorname{gcd}\\{a, b\\} .\) Then \(a / d\) and \(b / d\) are relatively prime.

Verify each. $$3+\lg n=\Omega(\lg n)$$

Let \(a, b, c,\) and \(n\) be any positive integers and \(p\) be any prime. Prove each. \(\operatorname{gcd}\\{n a, n b\\}=n \cdot \operatorname{gcd}\\{a, b\\}\)

Let \(a, b, c,\) and \(n\) be any positive integers and \(p\) be any prime. Prove each. \(\operatorname{gcd}\\{\operatorname{gcd}\\{a, b\\}, c\\}=\operatorname{gcd}\\{a, \operatorname{gcd}\\{b, c\\}\\}\)

Let \(a_{n}\) denote the number of times the statement \(x \leftarrow x+1\) is executed in the following loop: for \(i=1\) to \(n\) do for \(j=1\) to \(\lfloor i / 2\rfloor d o\) \(x \leftarrow x+1\) Show that $$a_{n}=\left\\{\begin{array}{ll} \frac{n^{2}}{4} & \text { if } n \text { is even } \\ \frac{n^{2}-1}{4} & \text { if } n \text { is odd } \end{array}\right.$$

Verify each. $$3 \lg n+2=\Omega(\lg n)$$

Let \(a, b, c,\) and \(n\) be any positive integers and \(p\) be any prime. Prove each. Let \(a | c\) and \(b | c,\) where \(a\) and \(b\) are relatively prime numbers. Then \(a b | c\).

Find the number of trailing zeros in the decimal value of each. $$100 !$$

Find the number of trailing zeros in the decimal value of each. $$378 !$$

Let \(a, b, c,\) and \(n\) be any positive integers and \(p\) be any prime. Prove each. 2 and 3 are the only two consecutive integers that are primes.