Joni Kenny

2021-01-05

Let a,b be coprime integers. Prove that every integer

Mayme

Skilled2021-01-06Added 103 answers

We work under the given condtions, a, b positive coprime integers, gcd

Example:

The following are representable in the required form 3,6,9,10,12,13,15,16

The following are not representable in the required form 1,2,4,5,7,8,11,14,17.

But, as per theorm. all

Describing all integral solutions (no conditions) of the equation

Given: a,b positive integers,

By Euclidean algorithm,

Consider any general solution

But

So,

Thus, if

Note, that the m's can be brought to {0,1,2,...,a-1} In particular, m are all non-negative

We have shown that the largest x for which the representation

Now, note that

Thus, the set of all x not admitting non-negative solutions for

The largest such x corresponds to

We have already proved that x=ab-a-b is not representable in the required form. Here is another proof.

Now, the equation

So, the general solution is

But

So

How to find out the mirror image of a point?

Generators of a free group

If G is a free group generated by n elements, is it possible to find an isomorphism of G with a free group generated by n-1 (or any fewer number) of elements?How many 3/4 Are in 1

Convert 10 meters to feet. Round your answer to the nearest tenth

6. Reduce the following matrix to reduced row echelon form:

Let v be a vector over a field F with zero vector 0 and let s,T be a substance of V .then which of the following statements are false

Describe Aut(Zp), the automorphism group of the cyclic group Zp where p is prime. In particular find the order of this group. (Hint: A generator must map to another generator)