Download Math 490, Homework #1

Math 490, Homework #1
Problems from text Section 2.3: 2, 5-14, 17, 18
2. Without converting to decimals, what is the least positive integer B for which
"
B
'%! œ "!C ?
5. If the number 412 is added to 3,2 and the result is divisible by 9, tell what the value of
, is.
6. (Question from the Classroom) A student claims to have made a discovery that, if you
take any two odd numbers, 7 and 8, then the differenceof their squares is divisible by 8.
She shows the example *#  &# œ &' which is divisible by 8 amd claims it is always true.
Is she correct? Prove or disprove this.
7. Investigate on the Internet a test for divisibility by 7. Explain what makes it complex.
Try to explain why and/or prove the test works.
8. State a test for divisibility by 8. Prove it works.
9. State a test for divisibility by 10. Prove it works.
10. What is the smallest positive integer composed of only even digits that is divisible by
9? Justify your answer.
11. Show that, if a number R is divisible by + and ,, and + and , have no common factor
other than 1, then R is divisible by +,.
12. Suppose that B and C are integers and that #B  $C is a multiple of 17. Show that
*B  &C is also a multiple of 17. [Hint: Start with "(B  "(C.]
13. How many numbers less than 1000 are divisible by either 5 or 7? Justify your
answer.
14. Are there single digit values for + and , that make the number 4324+5,4 divisible by
both 4 and *? If so, what are they? If not, why not?
17. (Question from the Classroom) After playing around with her calculator, a student
notices that the following numbers are all divisible by 11: (a) 123,123; (b) 742,742; (c)
685,685. She is convinced that the number +,- ,+,- where +, ,, and - are single digit
natural numbers will always be divisible by 11. Prove her conjecture or find a
counterexample. (Here +,- does not mean the product of +, ,, and - , rather the digits in
the representation of the number.)
18. Let +# œ "!!", +$ œ "!!"!!", +% œ "!!"!!"!!", and so on where the given pattern
continues and the subscript of + represents the number of 1's. Show that every single +8
is factorable if 8 is divisible by 3, or if 8 is even. (In fact, regardless of what 8 is, +8 is
factorable, but this is harder to prove.)