1) Let a and b be natural numbers. PROVE If a > b, then a does not divide b. 2)
ID: 2943525 • Letter: 1
Question
1) Let a and b be natural numbers. PROVE If a > b, then a does not divide b.
2) CONJECTURE Let a, b, and c be integers. PROVE If a divides (b+c), then a divides b or a divides c.
Explanation / Answer
1) If a were to divide b, then am=b for some natural number m. But, for all natural numbers a and m, the a*m>a. And, since a>b, we have a*m>a>b, so a*m>b. If a*m is greater than b, then a*m does not equal b. So, no such number m exists, so a does not divide b. 2) The conjecture is false. A counterexample: a = 5, b = 12, c=13. Then b+c=25. So, a divides b+c (5 divides 25), but a doesn't divide b or c (5 doesn't divide 12 or 13). Since the statement is false, it's impossible to prove that it's true! :)
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.