# which number is larger?

#### sfspitfire23

Fellas, I have a question regarding quantitative comparisons.

Say we have, in column A, (78)(243) and in column B, (77)(244).

I'm asked to determine which is larger.

Is it a good strategy to pick out the first two numbers in each variable? So, in column A we would take (78)(24) and B would be (77)(24). Thus A is larger because 78>77.

What about A= (90,021)(100,210) and B=(90,210)(100,021)? Using my method, we have (900)(100) in A and (902)(100) in B. Thus, B is larger.

Is it possible to do this? Is there some law?

#### Ackbeet

MHF Hall of Honor
What does the notation (78)(243) mean? I mean, I normally think usual arithmetic product of the two numbers. But in number theory, you might have a different context.

#### sfspitfire23

Sorry, it is the product

#### Ackbeet

MHF Hall of Honor
Ok, so I would say that, given two numbers whose sum is a constant, their product is maximized when the two numbers are closer together (global max is when they are equal). So I would say instantly that the product in Column A is greater. You can prove this using calculus: assume $$\displaystyle x+y=\text{constant}=c.$$ The goal is to maximize the product $$\displaystyle xy=x(c-x).$$ So, let

$$\displaystyle f(x)=x(c-x).$$ Then

$$\displaystyle f'(x)=c-2x.$$ Setting this equal to zero implies

$$\displaystyle c-2x=0,$$ or $$\displaystyle x=c/2=y.$$ Done.

• undefined and sfspitfire23

#### undefined

MHF Hall of Honor
Fellas, I have a question regarding quantitative comparisons.

Say we have, in column A, (78)(243) and in column B, (77)(244).

I'm asked to determine which is larger.

Is it a good strategy to pick out the first two numbers in each variable? So, in column A we would take (78)(24) and B would be (77)(24). Thus A is larger because 78>77.

What about A= (90,021)(100,210) and B=(90,210)(100,021)? Using my method, we have (900)(100) in A and (902)(100) in B. Thus, B is larger.

Is it possible to do this? Is there some law?
(78)(243) = (77 + 1)(243) = 77 * 243 + 243

(77)(244) = (77)(243 + 1) = 77 * 243 + 77

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#### sfspitfire23

Very nice I get it, thanks!

#### Also sprach Zarathustra

Try to prove the following for a, b integers greater than 1.

a*b>(a-1)*(b+1)

• undefined

#### Ackbeet

MHF Hall of Honor
You're welcome. undefined and AsZ's methods are also both entirely valid. AsZ's method might be generalized by proving that

ab > (a-m)(b+m).

#### undefined

MHF Hall of Honor
Try to prove the following for a, b integers greater than 1.

a*b>(a-1)*(b+1)
You forgot to mention the restriction $$\displaystyle \displaystyle a \le b$$. (Edit: Possibly it was an intentional omission though.)

Last edited:
• Also sprach Zarathustra