Homework Statement
Write down the negations of the following assertions (where m, n, a, b are natural numbers):
a) if Coke is not worse than Pepsi then nothing Mandelson says can be trusted.
b) \forall m \exists n\forall a\forall b (n >= m) /\ [(a=1) \/ (b=1) \/ (ab \ne n)]
Answers:
a) if...
Homework Statement
Find the convergents to the fraction 57/44.
Homework Equations
Could anyone provide any good internet resuorces about "finding the convergents to the fraction"?
The Attempt at a Solution
Let's consider 3 possible cases:
1) a = 3k. This case is impossible, since from equality 10^n + 1 = a*a*c we notice that the remainder, when LHS is divided by 3, is always 2, and the remainder, when RHS is divided by 3, is 0;
2) a = 3k + 1. Then a*a = 9k*k + 6k + 1 - the remainder, when a...
Homework Statement
Prove that 10^n + 1 cannot be expressed in the form a*a*c, where n,a,c are positive integers.
Homework Equations
By considering the reminder of 10^n + 1 when it is devided by 3, I arrived at the conclusion that:
a*a = 3z +1;
c = 3v + 2
for some positive integers...