91Ó°ÊÓ

Find the prime factorizations of the following integers. (a) 105 (b) 414 (c) 168 (d) 1612 (e) 9177

The octal representation of an integer uses powers of 8 in place notation. The digits of an octal number run from 0 to 7 , one never sees 8 's or 9 's. How would you represent 8 and 9 as octal numbers? What octal number comes immediately after \(777_{8}\) ? What (decimal) number is \(777_{8}\) ?

Identify each as rational or irrational. (a) \(5021.2121212121 \ldots\) (b) \(0.2340000000 \ldots\) (c) \(12.31331133311133331111 \ldots\) (d) \(\pi\) (e) \(2.987654321987654321987654321 \ldots\)

One method of converting from decimal to some other base is called repeated division. One divides the number by the base and records the remainder \(-\) one then divides the quotient obtained by the base and records the remainder. Continue dividing the successive quotients by the base until the quotient is smaller than the base. Convert 3267 to base- 7 using repeated division. Check your answer by using the meaning of base- 7 place notation. (For example \(54321_{7}\) means \(5 \cdot 7^{4}+\) \(\left.4 \cdot 7^{3}+3 \cdot 7^{2}+2 \cdot 7^{1}+1 \cdot 7^{0} .\right)\)

Give a description of the set of rational numbers whose decimal expansions terminate. (Alternatively, you may think of their decimal expansions ending in an infinitely-long string of zeros.)

Find the first 20 decimal places of \(\pi, 3 / 7, \sqrt{2}, 2 / 5,16 / 17, \sqrt{3}, 1 / 2\) and \(42 / 100 .\) Classify each of these quantity's decimal expansion as: terminating, having a repeating pattern, or showing no discernible pattern.

Consider the process of long division. Does this algorithm give any in- sight as to why rational numbers have terminating or repeating decimal expansions? Explain.

Suppose that 340 pounds of sand must be placed into bags having a 50 pound capacity. Write an expression using either floor or ceiling notation for the number of bags required.