Question

Use proof by contradiction to prove that there doesnt exist a smallest positive rational number

Answer 1

What?

Answer 2

@ganeshie8

Answer 3

Hmm you wrote rational integer. This is obsolete, integers are always rational. What is the full question?

Answer 4

Thanks for notifying

Answer 5

@mathmate

Answer 6

I am aware of the basic idea that we have to assume a smallest number in form of a/b and show that a/2b is smaller than a/b thus contradicting the statement. My question is how should I provide it formally?

Answer 7

if x is the smallest rational number
x/2 is a smaller rational number

Answer 8

One more question
Verify that 5^5 =1(mod)11. Hence find the remainder obtained on dividing 5^2016 by 11

Answer 9

wait, is zero a rational number?

Answer 10

No

Answer 11

why not?

Answer 12

Yes yes it is rational sorry

Answer 13

does smallest mean negative numbers then?

Answer 14

No question says positive rational numbers

Answer 15

now it does.

Answer 16

Can you explain the second question

Answer 17

Verify that 5^5 =1(mod)11. Hence find the remainder obtained on dividing 5^2016 by 11

Answer 18

Better to show a/b < ka/b for all positive rational k (to keep ka/b rational) to generate the set of all positive rational numbers.

Define a/b to be the smallest rational positive number, the set of positive rationals is then generated by ka/b where k is a positive rational number.
Then we say a/b < ka/b as a/b is the smallest rational.
We can rewrite this as 1 < k.
In other words k > 1. However we defined the set k to be all positive rationals and thus k > 1 is inadequate to describe the set.

Answer 19

hint:
\(a^k~ mod~ n \equiv (a~ mod~ n)^k~ mod~ n\)

Answer 20

@mathmate Can you provide a proof for that, since I haven't been taught that equation in college?

Answer 21

use fermat little theorem for 5^2016 by 11

Answer 22

Can it not be solved using induction, series or summation because this question was from that chapter

Answer 23

If mathematical induction is your topic, then you can start by proving
(a mod n)(b mod n) \(\equiv\) ab mod n
(see for example
http://math.stackexchange.com/questions/104721/proving-properties-of-modular-arithmetic-by-induction
)
then decompose 5^2016 into 5*(5^5)^403
and proceed with induction proof to show that 5^2016 \(\equiv\) 5(5^5) \(\equiv\)...

Answer 24

or better show, use induction to show that
\(a^k~ mod~ n \equiv (a~ mod~ n)^k~ mod~ n\)