Q1. For each of the following values of n and d find integers q and r such that n =dq+r and 0≤r
n = 70, d= 8
n= 62, d=7
n=-45, d=10
n=3, d=7
n=36, d=6
Q2. Prove that for all real numbers x and y ,|x+y|≤|x|+|y| (the triangle inequality)
Q3. Prove that for all integers n,n^2-n +5 is odd.
Q4. Give reason for your answer in the following statements. Assume that m and n variables are integer:
Is 52 divisible by 13?
Does 6|0
Is 6n(4n+10) divisible by 4
Is 2m(5m+15)divisible by 5m
Q5. Suppose a is an integer. If a mod 12 is 5 , what is 8a mod 12:
Q6. Determine whether the statement below is true or false. Prove the statement directly from definition if it is true, and give a counterexample if it is false:
For all integers a,b, and c, if a|(b+c), then a|b or a|c
Q7. Is it possible to have 50 coins, made up of pennies, dimes, and quarters, that add up to $3? Explain.
Q8. Prove that ?2 is irrational