problems 17b and 19c 17. Let be an integer (a) Show that n is not divisible by 3

Question

problems 17b and 19c

17. Let be an integer (a) Show that n is not divisible by 3 if and only if n2 3n 2 is a multiple of 3. (b) Show that n is a multiple of 3 if and only if n2 - 3n + 2 is not divisible by 3. 18. Give a formal proof of q r from the hypotheses p(q r) and q p. 19, The logical equivalence p (q Vr) (p ^--q) r of Exercise 7 is often used to prove p (q V r) by proving (p^-q) r instead. Use this idea to prove the following propositions about positive integers m and n (a) If m n 2 8, then m 3 or n 2 4. (b)If 2 25, then m 3 or n 2 5. (c) If m* +n 25 and n is a multiple of 3, then m 2 4 or n 6.

Explanation / Answer

17.b

if n is not divisible by 3 then
n = 3k + 1 or n = 3k +2
in both cases
n² - 3n + 2 is a multiple of 3

(3k + 1)² - 3(3k + 1) + 2 = 9k² - 3k = 3k(3k - 1)

(3k + 2)² - 3(3k + 2) + 2 = 9k² + 3k = 3k(3k + 1)


if n² - 3n + 2 is a multiple of 3,

then n² - 3n + 2 = 3k (*)

if n was divisible by 3, then
n = 3h

substitute in (*)
9h² - 9h + 2 = 3k

hence
3k - 9h² - 9h = 2
3(k - 3h² - 3h) = 2

k - 3h² - 3h = 2/3
which is impossible because h and k are integers

therefore n is NOT divisible by 3

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