What position does each man fill? Submit a listing and a run of your program. (Note: For your reference, see the Bank Employee example in Lecture 11 and the code lecture10andil.py) 3. (16pts) Variation on Ch4-4.4 #16 Prove that for all integers a, b,c,d and a= 0, b = 0, d = 0, if ab, djc, b|d, then a? |(11c2d-2ab + b3 ). Prove statements 15 and 16 directly from the definition of divisibility 15. For all integers a, b, and c. if a band ale then al(b+c). Answer + Proof: Suppose a, b, and care any integers such that a band al. [We must show that a|(+c). By definition of divides, b = ar and eas for some integers r and s. Then b+c= ar + as = als) by algebra. Let t=r+s. Thent is an integer (being a sum of integers), and thus b + c = at where t is an integer. By definition of divides, then, a|(b + c) (as was to be shown]. 16. For all integers a, b, and c. if a b and a c then a|(b c). Answer + Hint: The conclusion to be shown is that a certain quantity is divisible by a. To show this, you need to show that the quantity equals a times some integer. 17. For all integers a, b, c, and d, if alcand bld then abled. 18. Consider the following statement: The negative of any multiple of 3 is a multiple of 3. a. Write the statement formally using a quantifier and a variable. Answer integers n if n is a multiple on then -n is a multiple of 3. b. Determine whether the statement is true or false and justify your answer. Answer + The statement is true. Proof: Suppose n is any integer that is a multiple of 3. [We must show that -n is a multiple of 3.] By definition of multiple, n = 3k for some integer k. Then -n - (3) by substitution =3(-5) by algebra. Now -k is an integer because k is. Hence, by definition of multiple, -n is a multiple of 3 (as was to be shown] What position does each man fill? Submit a listing and a run of your program. (Note: For your reference, see the Bank Employee example in Lecture 11 and the code lecture10andil.py) 3. (16pts) Variation on Ch4-4.4 #16 Prove that for all integers a, b,c,d and a= 0, b = 0, d = 0, if ab, djc, b|d, then a? |(11c2d-2ab + b3 ). Prove statements 15 and 16 directly from the definition of divisibility 15. For all integers a, b, and c. if a band ale then al(b+c). Answer + Proof: Suppose a, b, and care any integers such that a band al. [We must show that a|(+c). By definition of divides, b = ar and eas for some integers r and s. Then b+c= ar + as = als) by algebra. Let t=r+s. Thent is an integer (being a sum of integers), and thus b + c = at where t is an integer. By definition of divides, then, a|(b + c) (as was to be shown]. 16. For all integers a, b, and c. if a b and a c then a|(b c). Answer + Hint: The conclusion to be shown is that a certain quantity is divisible by a. To show this, you need to show that the quantity equals a times some integer. 17. For all integers a, b, c, and d, if alcand bld then abled. 18. Consider the following statement: The negative of any multiple of 3 is a multiple of 3. a. Write the statement formally using a quantifier and a variable. Answer integers n if n is a multiple on then -n is a multiple of 3. b. Determine whether the statement is true or false and justify your answer. Answer + The statement is true. Proof: Suppose n is any integer that is a multiple of 3. [We must show that -n is a multiple of 3.] By definition of multiple, n = 3k for some integer k. Then -n - (3) by substitution =3(-5) by algebra. Now -k is an integer because k is. Hence, by definition of multiple, -n is a multiple of 3 (as was to be shown]