Divisibility Homework

Directions: Write legibly and in pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution, show all steps, and write the final answer in a sentence. Do not turn in your scratch work. Staple your pages together, in the correct order, and use this page as a cover sheet.

(1) Show that is divisible by 5.

(2) Show that 888 is divisible by 0.

(3) Show that is divisible by 7.

(4) Show that is divisible by 13.

(3) Which of the following integers is divisible by 7: 0, 123321, 707, , and

(4) Decide which integer is divisible by 22: 192544, 444, and

(5) Find the quotient and remainder in the division algorithm, (a) with divisor 17 and dividend (b) with divisor 17 and dividend

(6) Find the quotient and remainder in the division algorithm, with divisor 17 and dividend 289.

(7) What can you conclude if and are nonzero integers such that and

(8) Are there integers and such that but and

(9) Show that if and are positive integers and then

(10) Show that if is an integer, then divides

(11) Show that the square of every of odd integer is of the form

(12) Show that the product of every two integers of the form is of the form

(13) Use mathematical induction to show that is divisible by 5 for every positive integer

(14) Use mathematical induction to show that the sum of the cubes of three consecutive integers is divisible by 9.

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