Question

Prove that if n = ab, where a and b are positive integers, then \(a \le \sqrt n\) or \(b \le \sqrt n\)

Answer 1

prove by contrapositive

Answer 2

hmm....and that means...?

Answer 3

The contrapositive would be:
If \(a > \sqrt{n} and b > \sqrt{n}\) then \(n \ne ab\).

Answer 4

show that \[a >\sqrt{n}\] and \[b>\sqrt{n}\] implies \[n \neq ab\]

Answer 5

hmm 3 posts....

Answer 6

Shhh, there's no edit button.

Answer 7

And for some reason, no preview. When did that happen?

Answer 8

okay... so...

\[a^2 > n\]
\[b^2 > n\]

then....

\[a^2 - b^2 = 0\]

Answer 9

what am i doing with my life...

Answer 10

How do you get \(a^2-b^2 = 0\)? That's not right.

Answer 11

\[a^2 - b^2 > 0\]
you know what i mean\t

Answer 12

Still not right.

Answer 13

??????!!!!!!!!!!

Answer 14

assume that \[a>\sqrt{n}\] and \[b>\sqrt{n}\]

what do you get for ab ?

Answer 15

b could be larger than a. That is not restricted by your assumption, nor does it need to be.

Answer 16

fine.

\[a > \sqrt n\]
\[b > \sqrt n\]
\[a - b > 0\]

happy?

Answer 17

Not super happy. I'm not sure why you care so much about the difference of a and b, when the proof is mostly concerned with their product.

Answer 18

.........

Answer 19

your goal is to show that \[n \neq ab\]
why do you subtract a-b?

Answer 20

because i can...

Answer 21

lol okay.

Answer 22

.....i hate math.....

Answer 23

now try to multiply a and b, what do you get? :D

Answer 24

if a > sqrt n and b > sqrt n.... then \[ab > \sqrt n \cdot \sqrt n\]
\[ab > n\]

Answer 25

is that allowed?

Answer 26

That's the basic logic of it.

Answer 27

yes..., :D

Answer 28

You start with
\(a > \sqrt{n}\)
Multiply both sides of the equation by b. Do we have to change the direction of the inequality? No, because we are assuming that b is positive.
\(ab > \sqrt{n}b\)

Answer 29

Now, because \(b > \sqrt{n}\)
\(\sqrt{n}b > \sqrt{n}\sqrt{n} = n\)

Answer 30

Therefore, ab > n.

Answer 31

that's really allowed?

Answer 32

Kind of. You have to do those middle steps, and it's important to understand that multiplying both sides by b does not change the direction of the inequality only because we are assuming b is positive.

Answer 33

Allow me to demonstrate.

2 > 1

Shall I multiply both sides by x? Sure, let's do it.

2x > x
Seems true, right? Well, it's true if x is positive.
x=5
10 > 5
What about if x is negative?
x=-5
-10 > -5
Aw sadness. Not true anymore.